A Tutorial on the Classical Theories of Electromagnetic Scattering and Diffraction
11 A Tutorial on the Classical Theories of Electromagnetic Scattering and Diffraction
Masud Mansuripur
College of Optical Sciences, The University of Arizona, Tucson [Published in
Nanophotonics , eISSN 2192-8614, ISSN 2192-8606, doi: 10.1515/nanoph-2020-0348 (2020)]
Abstract . Starting with Maxwell’s equations, we derive the fundamental results of the Huygens-Fresnel-Kirchhoff and Rayleigh-Sommerfeld theories of scalar diffraction and scattering. These results are then extended to cover the case of vector electromagnetic fields. The famous Sommerfeld solution to the problem of diffraction from a perfectly conducting half-plane is elaborated. Far-field scattering of plane waves from obstacles is treated in some detail, and the well-known optical cross-section theorem, which relates the scattering cross-section of an obstacle to its forward scattering amplitude, is derived. Also examined is the case of scattering from mild inhomogeneities within an otherwise homogeneous medium, where, in the first Born approximation, a fairly simple formula is found to relate the far-field scattering amplitude to the host medium’s optical properties. The related problem of neutron scattering from ferromagnetic materials is treated in the final section of the paper.
1. Introduction . The classical theories of electromagnetic (EM) scattering and diffraction were developed throughout the nineteenth century by the likes of Augustine Jean Fresnel (1788-1827), Gustav Kirchhoff (1824-1887), John William Strutt (Lord Rayleigh, 1842-1919), and Arnold Sommerfeld (1868-1951).
A thorough appreciation of these theories requires an understanding of the Maxwell-Lorentz electrodynamics and a working knowledge of vector calculus, differential equations, Fourier transformation, and complex-plane integration techniques. The relevant physical and mathematical arguments have been covered (to varying degrees of clarity and completeness) in numerous textbooks, monographs, and research papers.
The goal of this tutorial is to present the core concepts of the classical theories of scattering and diffraction by starting with Maxwell’s equations and deriving the fundamental results using mathematical arguments that should be accessible to students of optical sciences as well as practitioners of modern optical engineering and photonics technologies. A consistent notation and uniform terminology is used throughout the paper. To maintain the focus on the main results and reduce the potential for distraction, some of the longer derivations and secondary arguments have been relegated to the appendices. The organization of the paper is as follows. After a brief review of Maxwell’s equations in Sec.2, we provide a detailed analysis of an all-important Green function in Sec.3. The Huygens-Fresnel-Kirchhoff scalar theory of diffraction is the subject of Sec.4, followed by the Rayleigh-Sommerfeld modification and enhancement of that theory in Sec.5. These scalar theories are subsequently generalized in Sec.6 to arrive at a number of formulas for vector scattering and vector diffraction of EM waves under various settings and circumstances. Section 6 also contains a few examples that demonstrate the application of vector diffraction formulas in situations of practical interest. The famous Sommerfeld solution to the problem of diffraction from a perfectly electrically conducting half-plane is presented in some detail in Sec.7. Applying the vector formulas of Sec.6 to far-field scattering, we show in Sec.8 how the forward scattering amplitude for a plane-wave that illuminates an arbitrary object relates to the scattering cross-section of that object. This important result in the classical theory of scattering is formally known as the optical cross-section theorem (or the optical theorem).
An alternative approach to the problem of EM scattering when the host medium contains a region of weak inhomogeneities is described in Sec.9. Here, we use Maxwell’s macroscopic equations in conjunction with the Green function of Sec.3 to derive a fairly simple formula for the far-field scattering amplitude in the first Born approximation. The related problem of slow neutron scattering from ferromagnetic media is treated in Sec.10. The paper closes with a few conclusions and final remarks in Sec.11.
2. Maxwell’s equations . The standard equations of the classical Maxwell-Lorenz theory of electrodynamics relate four material sources to four EM fields in the Minkowski spacetime ( 𝒓𝒓 , 𝑡𝑡 ) . The sources are the free charge density 𝜌𝜌 free , free current density 𝑱𝑱 free , polarization 𝑷𝑷 , and magnetization 𝑴𝑴 , while the fields are the electric field 𝑬𝑬 , magnetic field 𝑯𝑯 , displacement 𝑫𝑫 , and magnetic induction 𝑩𝑩 . In the 𝑆𝑆𝑆𝑆 system of units, where the free space (or vacuum) has permittivity 𝜀𝜀 and permeability 𝜇𝜇 , the displacement is defined as 𝑫𝑫 = 𝜀𝜀 𝑬𝑬 + 𝑷𝑷 , and the magnetic induction as 𝑩𝑩 = 𝜇𝜇 𝑯𝑯 + 𝑴𝑴 . Let the total charge-density 𝜌𝜌 ( 𝒓𝒓 , 𝑡𝑡 ) and total current-density 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) be defined as 𝜌𝜌 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜌𝜌 free ( 𝒓𝒓 , 𝑡𝑡 ) − 𝜵𝜵 ∙ 𝑷𝑷 ( 𝒓𝒓 , 𝑡𝑡 ) . (1) 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑱𝑱 free ( 𝒓𝒓 , 𝑡𝑡 ) + 𝜕𝜕 𝑡𝑡 𝑷𝑷 ( 𝒓𝒓 , 𝑡𝑡 ) + 𝜇𝜇 −1 𝜵𝜵 × 𝑴𝑴 ( 𝒓𝒓 , 𝑡𝑡 ) . (2) The charge-current continuity equation, 𝜵𝜵 ∙ 𝑱𝑱 + 𝜕𝜕 𝑡𝑡 𝜌𝜌 = 0 , is generally satisfied by the above densities, irrespective of whether their corresponding sources are free (i.e., 𝜌𝜌 free and 𝑱𝑱 free ), or bound electric charges within electric dipoles (i.e., −𝜵𝜵 ∙ 𝑷𝑷 and 𝜕𝜕 𝑡𝑡 𝑷𝑷 ), or bound electric currents within magnetic dipoles (i.e., 𝜇𝜇 −1 𝜵𝜵 × 𝑴𝑴 ). Invoking Eqs.(1) and (2), Maxwell’s macroscopic equations are written as follows: 𝜀𝜀 𝜵𝜵 ∙ 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜌𝜌 ( 𝒓𝒓 , 𝑡𝑡 ) . (3) 𝜵𝜵 × 𝑩𝑩 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜇𝜇 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) + 𝜇𝜇 𝜀𝜀 𝜕𝜕 𝑡𝑡 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) . (4) 𝜵𝜵 × 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = −𝜕𝜕 𝑡𝑡 𝑩𝑩 ( 𝒓𝒓 , 𝑡𝑡 ) . (5) 𝜵𝜵 ∙ 𝑩𝑩 ( 𝒓𝒓 , 𝑡𝑡 ) = 0 . (6) Taking the curl of both sides of the third equation, using the vector identity 𝜵𝜵 × 𝜵𝜵 × 𝑽𝑽 = 𝜵𝜵 ( 𝜵𝜵 ∙ 𝑽𝑽 ) − 𝜵𝜵 𝑽𝑽 , and substituting from the first and second equations, we find 𝜵𝜵 × 𝜵𝜵 × 𝑬𝑬 = −𝜇𝜇 𝜕𝜕 𝑡𝑡 𝑱𝑱 − 𝜇𝜇 𝜀𝜀 𝜕𝜕 𝑡𝑡2 𝑬𝑬 → ( 𝜵𝜵 − 𝑐𝑐 −2 𝜕𝜕 𝑡𝑡2 ) 𝑬𝑬 = 𝜇𝜇 𝜕𝜕 𝑡𝑡 𝑱𝑱 + 𝜀𝜀 −1 𝜵𝜵𝜌𝜌 . (7) Here, 𝑐𝑐 = ( 𝜇𝜇 𝜀𝜀 ) − ½ is the speed of light in vacuum. Similarly, taking the curl of both sides of Eq.(4) and substituting from Eqs.(5) and (6), we find 𝜵𝜵 × 𝜵𝜵 × 𝑩𝑩 = 𝜇𝜇 𝜵𝜵 × 𝑱𝑱 − 𝜇𝜇 𝜀𝜀 𝜕𝜕 𝑡𝑡2 𝑩𝑩 → ( 𝜵𝜵 − 𝑐𝑐 −2 𝜕𝜕 𝑡𝑡2 ) 𝑩𝑩 = −𝜇𝜇 𝜵𝜵 × 𝑱𝑱 . (8) The scalar and vector potentials, 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) and 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) , are defined such that 𝑩𝑩 = 𝜵𝜵 × 𝑨𝑨 and 𝑬𝑬 = −𝜵𝜵𝜓𝜓 − 𝜕𝜕 𝑡𝑡 𝑨𝑨 . With these definitions, Maxwell’s third and fourth equations are automatically satisfied. In the Lorenz gauge, where 𝜵𝜵 ∙ 𝑨𝑨 + 𝑐𝑐 −2 𝜕𝜕 𝑡𝑡 𝜓𝜓 = 0 , Maxwell’s second equation yields 𝜵𝜵 × 𝜵𝜵 × 𝑨𝑨 = 𝜇𝜇 𝑱𝑱 − 𝜇𝜇 𝜀𝜀 𝜵𝜵 ( 𝜕𝜕 𝑡𝑡 𝜓𝜓 ) − 𝜇𝜇 𝜀𝜀 𝜕𝜕 𝑡𝑡2 𝑨𝑨 → ( 𝜵𝜵 − 𝑐𝑐 −2 𝜕𝜕 𝑡𝑡2 ) 𝑨𝑨 = −𝜇𝜇 𝑱𝑱 . (9) Similarly, substitution into Maxwell’s first equation for the 𝐸𝐸 -field in terms of the potentials yields −𝜀𝜀 𝜵𝜵 ∙ ( 𝜵𝜵𝜓𝜓 + 𝜕𝜕 𝑡𝑡 𝑨𝑨 ) = 𝜌𝜌 → ( 𝛻𝛻 − 𝑐𝑐 −2 𝜕𝜕 𝑡𝑡2 ) 𝜓𝜓 = − 𝜌𝜌 𝜀𝜀 ⁄ . (10) Since monochromatic fields oscillate at a single frequency 𝜔𝜔 , their time-dependence factor is generally written as exp( − i 𝜔𝜔𝑡𝑡 ) . Consequently, the spatiotemporal dependence of all the fields and all the sources can be separated into a space part and a time part. For example, the 𝐸𝐸 -field may now be written as 𝑬𝑬 ( 𝒓𝒓 ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 , the total electric charge-density as 𝜌𝜌 ( 𝒓𝒓 ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 , and so on. Defining the free-space wavenumber 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ , the Helmholtz equations (7) - (10) now become ( 𝜵𝜵 + 𝑘𝑘 ) 𝑬𝑬 ( 𝒓𝒓 ) = − i 𝜔𝜔𝜇𝜇 𝑱𝑱 ( 𝒓𝒓 ) + 𝜀𝜀 −1 𝜵𝜵𝜌𝜌 ( 𝒓𝒓 ) . (11) ( 𝜵𝜵 + 𝑘𝑘 ) 𝑩𝑩 ( 𝒓𝒓 ) = −𝜇𝜇 𝜵𝜵 × 𝑱𝑱 ( 𝒓𝒓 ) . (12) ( 𝜵𝜵 + 𝑘𝑘 ) 𝑨𝑨 ( 𝒓𝒓 ) = −𝜇𝜇 𝑱𝑱 ( 𝒓𝒓 ) . (13) ( 𝛻𝛻 + 𝑘𝑘 ) 𝜓𝜓 ( 𝒓𝒓 ) = − 𝜌𝜌 ( 𝒓𝒓 ) 𝜀𝜀 ⁄ . (14) In regions of free space, where 𝜌𝜌 ( 𝒓𝒓 ) = 0 and 𝑱𝑱 ( 𝒓𝒓 ) = 0 , the right-hand sides of Eqs.(11) -(14) vanish, thus allowing one to replace 𝑘𝑘 𝑬𝑬 ( 𝒓𝒓 ) with −𝜵𝜵 𝑬𝑬 ( 𝒓𝒓 ) , and similarly for 𝑩𝑩 ( 𝒓𝒓 ) , 𝑨𝑨 ( 𝒓𝒓 ) , and 𝜓𝜓 ( 𝒓𝒓 ) , whenever the need arises. These substitutions will be used in the following sections.
3. The Green function . In the spherical coordinate system ( 𝑟𝑟 , 𝜃𝜃 , 𝜑𝜑 ) , the Laplacian of the spherically symmetric function 𝐺𝐺 ( 𝒓𝒓 ) = 𝑒𝑒 i𝑘𝑘 𝑟𝑟 𝑟𝑟⁄ equals 𝜕𝜕 ( 𝑟𝑟𝐺𝐺 ) 𝑟𝑟𝜕𝜕𝑟𝑟 ⁄ = −𝑘𝑘 𝐺𝐺 ( 𝒓𝒓 ) everywhere except at the origin 𝑟𝑟 = 0 , where the function has a singularity. Thus, ( 𝛻𝛻 + 𝑘𝑘 ) 𝐺𝐺 ( 𝒓𝒓 ) = 0 at all points 𝒓𝒓 except at the origin. A good way to handle the singularity at 𝑟𝑟 = 0 is to treat 𝐺𝐺 ( 𝒓𝒓 ) as the limiting form of another function that has no such singularity, namely, 𝐺𝐺 ( 𝒓𝒓 ) = 𝑙𝑙𝑙𝑙𝑙𝑙 𝜀𝜀→0 �𝑒𝑒 i𝑘𝑘 𝑟𝑟 √𝑟𝑟 + 𝜀𝜀⁄ � . (15) The Laplacian of our well-behaved, non-singular function is readily found to be 𝑟𝑟 −2 𝜕𝜕 𝑟𝑟 �𝑟𝑟 𝜕𝜕 𝑟𝑟 �𝑒𝑒 i𝑘𝑘 𝑟𝑟 √𝑟𝑟 + 𝜀𝜀⁄ �� = − � ( 𝑟𝑟 +𝜀𝜀 ) − 𝜀𝜀𝑟𝑟 ( 𝑟𝑟 +𝜀𝜀 ) + 𝑘𝑘 ( 𝑟𝑟 +𝜀𝜀 ) � 𝑒𝑒 i𝑘𝑘 𝑟𝑟 . (16) The first two functions appearing inside the square brackets on the right-hand side of Eq.(16) are confined to the vicinity of the origin at 𝑟𝑟 = 0 ; they are tall, narrow, symmetric, and have the following volume integrals (see Appendix A for details): ∫ 𝜋𝜋𝑟𝑟 ( 𝑟𝑟 + 𝜀𝜀 ) −5 2⁄ d 𝑟𝑟 ∞0 = 4 𝜋𝜋 𝜀𝜀⁄ . (17) ∫ 𝜋𝜋𝑟𝑟 ( 𝑟𝑟 + 𝜀𝜀 ) −3 2⁄ d 𝑟𝑟 ∞0 = 4 𝜋𝜋 √𝜀𝜀⁄ . (18) Thus, in the limit of sufficiently small 𝜀𝜀 , the first two functions appearing on the right-hand side of Eq.(16) can be represented by 𝛿𝛿 -functions, † and the entire equation may be written as ( 𝛻𝛻 + 𝑘𝑘 )( 𝑒𝑒 i𝑘𝑘 𝑟𝑟 √𝑟𝑟 + 𝜀𝜀⁄ ) = − 𝜋𝜋 (1 − i2 𝑘𝑘 √𝜀𝜀 ) 𝛿𝛿 ( 𝒓𝒓 ) . (19) This is the sense in which we can now state that ( 𝛻𝛻 + 𝑘𝑘 ) 𝐺𝐺 ( 𝒓𝒓 ) = − 𝜋𝜋𝛿𝛿 ( 𝒓𝒓 ) in the limit when 𝜀𝜀 → . Shifting the center of the function to an arbitrary point 𝒓𝒓 , we will have ( 𝛻𝛻 + 𝑘𝑘 ) 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) = − 𝜋𝜋𝛿𝛿 ( 𝒓𝒓 − 𝒓𝒓 ) . (20) The gradient of the Green function 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) , which plays an important role in our discussions of the following sections, is now found to be 𝜵𝜵𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) = 𝜵𝜵 � 𝑒𝑒 i𝑘𝑘0 | 𝒓𝒓−𝒓𝒓0 | | 𝒓𝒓 − 𝒓𝒓 | � = (i 𝑘𝑘 − | 𝒓𝒓 − 𝒓𝒓 | −1 ) � 𝑒𝑒 i𝑘𝑘0 | 𝒓𝒓−𝒓𝒓0 | | 𝒓𝒓 − 𝒓𝒓 | � 𝒓𝒓 − 𝒓𝒓 | 𝒓𝒓 − 𝒓𝒓 | . (21) Appendix B provides an analysis of Eq.(20), an inhomogeneous Helmholtz equation, via Fourier transformation. † When multiplied by 𝜀𝜀 , as required by Eq.(16), the integrand in Eq.(17) peaks at ~ 𝜀𝜀 − ½ at 𝑟𝑟 = √⅔𝜀𝜀 , then drops steadily to ~ √𝜀𝜀 at 𝑟𝑟 = √𝜀𝜀 . Similarly, the integrand in Eq.(18), again multiplied by 𝜀𝜀 , peaks at ~1 at 𝑟𝑟 = √ ½ 𝜀𝜀 , then drops steadily to ~ √𝜀𝜀 at 𝑟𝑟 = √𝜀𝜀 .
4. The Huygens-Fresnel-Kirchhoff theory of diffraction . Consider a scalar function 𝜓𝜓 ( 𝒓𝒓 ) that satisfies the homogeneous Helmholtz equation ( 𝛻𝛻 + 𝑘𝑘 ) 𝜓𝜓 ( 𝒓𝒓 ) = 0 everywhere within a volume 𝑉𝑉 of free space enclosed by a surface 𝑆𝑆 . Two examples of the geometry under consideration are depicted in Fig.1. In general, 𝜓𝜓 ( 𝒓𝒓 ) can represent the scalar potential or any Cartesian component of the monochromatic 𝐸𝐸 -field, 𝐵𝐵 -field, or 𝐴𝐴 -field associated with an EM wave propagating in free space with frequency 𝜔𝜔 and wave-number 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ . In Fig.1(a), the EM wave arrives at the surface 𝑆𝑆 from sources located on the left-hand side of 𝑆𝑆 , and enters the volume 𝑉𝑉 contained within the closed surface 𝑆𝑆 = 𝑆𝑆 + 𝑆𝑆 . In Fig.1(b), the EM waves emanate from inside the closed surface 𝑆𝑆 and permeate the volume 𝑉𝑉 enclosed by 𝑆𝑆 on one side and by a second closed surface 𝑆𝑆 that defines the outer boundary of 𝑉𝑉 . In both figures, the point 𝒓𝒓 , where the field is being observed, is located inside the volume 𝑉𝑉 , and the surface normals 𝒏𝒏� everywhere on the closed surface 𝑆𝑆 are unit-vectors that point inward (i.e., into the volume 𝑉𝑉 ). Fig.1 . Two surfaces 𝑆𝑆 and 𝑆𝑆 bound the scattering region, which is assumed to be free of sources and material bodies. All the radiation within the scattering region comes from the outside. In (a) the sources of radiation are on the left-hand side of 𝑆𝑆 , while in (b) the radiation emanates from the sources inside the closed surface 𝑆𝑆 . The observation point 𝒓𝒓 is an arbitrary point within the scattering region. All the points located on 𝑆𝑆 are assumed to be far away from 𝒓𝒓 , so that the fields that reach 𝑆𝑆 do not contribute to the fields observed at 𝒓𝒓 . At each and every point on 𝑆𝑆 and 𝑆𝑆 , the surface normals 𝒏𝒏� point into the scattering region. The essence of the theory developed by G. Kirchhoff in 1882 (building upon the original ideas of Huygens and Fresnel) is an exact mathematical relation between the observed field 𝜓𝜓 ( 𝒓𝒓 ) and the field 𝜓𝜓 ( 𝒓𝒓 ) that exists everywhere on the closed surface 𝑆𝑆 . This relation is derived below using the sifting property of Dirac’s 𝛿𝛿 -function, the relation between the 𝛿𝛿 -function and the Green function given in Eq.(20), some well-known identities in standard vector calculus, and Gauss’s famous theorem of vector calculus, according to which ∫ 𝜵𝜵 ∙ 𝑽𝑽 d 𝑣𝑣 𝑉𝑉 = ∮ 𝑽𝑽 ∙ d 𝒔𝒔 𝑆𝑆 . We have 𝜓𝜓 ( 𝒓𝒓 ) = ∫ 𝜓𝜓 ( 𝒓𝒓 ) 𝛿𝛿 ( 𝒓𝒓 − 𝒓𝒓 )d 𝒓𝒓 𝑉𝑉 = − (4 𝜋𝜋 ) −1 ∫ 𝜓𝜓 ( 𝒓𝒓 )( 𝛻𝛻 + 𝑘𝑘 ) 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝒓𝒓 𝑉𝑉 = − (4 𝜋𝜋 ) −1 ∫ [ 𝜓𝜓 ( 𝒓𝒓 ) 𝜵𝜵 ∙ 𝜵𝜵𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) − 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) 𝜵𝜵 ∙ 𝜵𝜵𝜓𝜓 ( 𝒓𝒓 )]d 𝒓𝒓 𝑉𝑉 = − (4 𝜋𝜋 ) −1 ∫ [ 𝜵𝜵 ∙ ( 𝜓𝜓𝜵𝜵𝐺𝐺 ) − 𝜵𝜵𝐺𝐺 ∙ 𝜵𝜵𝜓𝜓 − 𝜵𝜵 ∙ ( 𝐺𝐺𝜵𝜵𝜓𝜓 ) + 𝜵𝜵𝜓𝜓 ∙ 𝜵𝜵𝐺𝐺 ]d 𝒓𝒓 𝑉𝑉 = (4 𝜋𝜋 ) −1 ∫ [ 𝜓𝜓 ( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) − 𝐺𝐺 ( 𝒏𝒏� ∙ 𝜵𝜵𝜓𝜓 )]d 𝑠𝑠 𝑆𝑆 = (4 𝜋𝜋 ) −1 ∫ [ 𝜓𝜓 ( 𝒓𝒓 ) 𝜕𝜕 𝑛𝑛 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) − 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) 𝜕𝜕 𝑛𝑛 𝜓𝜓 ( 𝒓𝒓 )]d 𝑠𝑠 𝑆𝑆 . (22) 𝜵𝜵 ∙ ( 𝜙𝜙𝑽𝑽 ) = 𝑽𝑽 ∙ 𝜵𝜵𝜙𝜙 + 𝜙𝜙𝜵𝜵 ∙ 𝑽𝑽 Replace 𝑘𝑘 𝜓𝜓 with −𝛻𝛻 𝜓𝜓 . 𝒏𝒏� 𝒓𝒓 𝑆𝑆 × 𝑆𝑆 𝒏𝒏� (a) 𝒏𝒏 � 𝒓𝒓 𝑆𝑆 × 𝑆𝑆 𝒏𝒏 � (b) Gauss’ theorem; 𝒏𝒏� points into the volume 𝑉𝑉 . See Eq.(20) Example . In the extreme situation where 𝑆𝑆 is a small sphere of radius 𝜀𝜀 centered at 𝒓𝒓 , we will have 𝜓𝜓 ( 𝒓𝒓 ) ≅ 𝜓𝜓 ( 𝒓𝒓 ) , 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) = 𝑒𝑒 i𝑘𝑘 𝜀𝜀 𝜀𝜀⁄ , and 𝜕𝜕 𝑛𝑛 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) = − (i 𝑘𝑘 − 𝜀𝜀 −1 ) 𝑒𝑒 i𝑘𝑘 𝜀𝜀 𝜀𝜀⁄ . Given that the surface area of the sphere is 𝜋𝜋𝜀𝜀 , the second term in Eq.(22) makes a negligible contribution to the overall integral when 𝜀𝜀 → . The first term, however, contains 𝑒𝑒 i𝑘𝑘 𝜀𝜀 𝜀𝜀 ⁄ , which integrates to 𝜋𝜋 in the limit of 𝜀𝜀 → , yielding 𝜓𝜓 ( 𝒓𝒓 ) as the final result. Taking the spherical (or hemi-spherical) surface 𝑆𝑆 in Fig.1 to be far away from the region of interest, the field 𝜓𝜓 ( 𝒓𝒓 ) everywhere on 𝑆𝑆 should have the general form of 𝑓𝑓 ( 𝜃𝜃 , 𝜑𝜑 ) 𝑒𝑒 i𝑘𝑘 𝑟𝑟 𝑟𝑟⁄ and, consequently, 𝜕𝜕 𝑛𝑛 𝜓𝜓 ( 𝒓𝒓 )~(i 𝑘𝑘 − 𝑟𝑟 −1 ) 𝜓𝜓 ( 𝒓𝒓 ) . Similarly, across the surface 𝑆𝑆 , the Green function has the asymptotic form 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )~ 𝑒𝑒 i𝑘𝑘 𝑟𝑟 𝑟𝑟⁄ and, therefore, 𝜕𝜕 𝑛𝑛 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )~(i 𝑘𝑘 − 𝑟𝑟 −1 ) 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) . Thus, on the faraway surface 𝑆𝑆 , the integrand in Eq.(22) must decline faster than 𝑟𝑟 ⁄ , which means that the contribution of 𝑆𝑆 to 𝜓𝜓 ( 𝒓𝒓 ) as given by Eq.(22) should be negligible. Kirchhoff’s diffraction integral now relates the field at the observation point 𝒓𝒓 to the field distribution on 𝑆𝑆 , as follows: 𝜓𝜓 ( 𝒓𝒓 ) = (4 𝜋𝜋 ) −1 ∫ [ 𝜓𝜓 ( 𝒓𝒓 ) 𝜕𝜕 𝑛𝑛 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) − 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) 𝜕𝜕 𝑛𝑛 𝜓𝜓 ( 𝒓𝒓 )]d 𝑠𝑠 𝑆𝑆1 . (23) To make contact with the Huygens-Fresnel theory of diffraction, Kirchhoff suggested that both 𝜓𝜓 ( 𝒓𝒓 ) and 𝜕𝜕 𝑛𝑛 𝜓𝜓 ( 𝒓𝒓 ) vanish on the opaque areas of the screen 𝑆𝑆 , whereas in the open (or transparent, or unobstructed) regions, they retain the profiles they would have had in the absence of the screen. These suggestions, while reasonable from a practical standpoint and resulting in good agreement with experimental observations under many circumstances, are subject to criticism for their mathematical inconsistency, as will be elaborated in the next section.
5. The Rayleigh-Sommerfeld theory . In the important special case where the surface 𝑆𝑆 coincides with the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 , one can adjust the Green function in such a way as to eliminate either the first or the second term in the integrand of Eq.(23). These situations would then correspond, respectively, to the so-called Neumann and Dirichlet boundary conditions. Let the observation point, which we assume to reside on the right-hand side of the planar surface 𝑆𝑆 , be denoted by 𝒓𝒓 + = ( 𝑥𝑥 , 𝑥𝑥 , 𝑧𝑧 ) , while its mirror image in 𝑆𝑆 (located on the left-hand side of 𝑆𝑆 ) is denoted by 𝒓𝒓 − = ( 𝑥𝑥 , 𝑥𝑥 , −𝑧𝑧 ) . If we use 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 + ) in Eq.(23), we obtain 𝜓𝜓 ( 𝒓𝒓 ) on the left-hand side of the equation, but if we use 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 − ) instead, the integral will yield zero — simply because the peak of the corresponding 𝛿𝛿 -function now resides outside the integration volume. Thus, we can replace 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) in Eq.(23) with either of the two functions 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 + ) ± 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 − ) . On the 𝑆𝑆 plane, where 𝑧𝑧 = 0 , we have 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 − ) = 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 + ) and 𝜕𝜕 𝑛𝑛 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 − ) = −𝜕𝜕 𝑛𝑛 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 + ) . The resulting diffraction integrals, respectively satisfying the Neumann and Dirichlet boundary conditions, will then be 𝜓𝜓 ( 𝒓𝒓 ) = − � exp ( i𝑘𝑘 | 𝒓𝒓−𝒓𝒓 |)| 𝒓𝒓−𝒓𝒓 | 𝜕𝜕 𝑛𝑛 𝜓𝜓 ( 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 . (24) 𝜓𝜓 ( 𝒓𝒓 ) = � ( i𝑘𝑘 − | 𝒓𝒓−𝒓𝒓 | −1 ) exp ( i𝑘𝑘 | 𝒓𝒓 − 𝒓𝒓 |)| 𝒓𝒓 − 𝒓𝒓 | [( 𝒓𝒓 − 𝒓𝒓 ) ∙ 𝒏𝒏� ] 𝜓𝜓 ( 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 . (25) Note that, while Eq.(23) applies to any arbitrary surface 𝑆𝑆 , the Rayleigh-Sommerfeld equations (24) and (25) are restricted to distributions that are specified on a flat plane. Given the scalar field profile 𝜓𝜓 ( 𝒓𝒓 ) and/or its gradient on a flat plane, all three equations are exact consequences of Maxwell’s equations. To apply these equations in practice, one must resort to some form of approximation to estimate the field distribution on 𝑆𝑆 . The conventional approximation is that, in the opaque regions of the screen 𝑆𝑆 , either 𝜓𝜓 ( 𝒓𝒓 ) or 𝜕𝜕 𝑛𝑛 𝜓𝜓 ( 𝒓𝒓 ) or both are vanishingly small and, therefore, negligible, whereas in the transparent (or unobstructed) regions of 𝑆𝑆 , the field 𝜓𝜓 ( 𝒓𝒓 ) and/or its gradient 𝜕𝜕 𝑛𝑛 𝜓𝜓 ( 𝒓𝒓 ) (along the surface-normal) retain the profile they would have had in the absence of the screen. In this way, one can proceed to evaluate the integral on the open (or transparent, or unobstructed) apertures of 𝑆𝑆 in order to arrive at a reasonable estimate of 𝜓𝜓 ( 𝒓𝒓 ) at the desired observation location. As a formula for computing diffraction patterns from one or more apertures in an otherwise opaque screen, the problem with Eq.(23) is that, when combined with Kirchhoff’s assumption that both 𝜓𝜓 and 𝜕𝜕 𝑛𝑛 𝜓𝜓 vanish on the opaque regions of the physical screen at 𝑆𝑆 , it becomes mathematically inconsistent. This is because an analytic function such as 𝜓𝜓 ( 𝒓𝒓 ) vanishes everywhere if both 𝜓𝜓 and 𝜕𝜕 𝑛𝑛 𝜓𝜓 happen to be zero on any patch of the surface 𝑆𝑆 . In contrast, Eq.(24), when applied to a physical screen, requires only the assumption that 𝜕𝜕 𝑛𝑛 𝜓𝜓 be zero on the opaque regions of the screen. While still an approximation, this is a much more mathematically palatable condition than the Kirchhoff requirement. Similarly, Eq.(25) requires only the approximation that 𝜓𝜓 be zero on the opaque regions. Thus, on the grounds of mathematical consistency, there is a preference for either Eq.(24) or Eq.(25) over Eq.(23). However, given the aforementioned approximate nature of the values chosen for 𝜓𝜓 and 𝜕𝜕 𝑛𝑛 𝜓𝜓 across the screen at 𝑆𝑆 , it turns out that scalar diffraction calculations based on these three formulas yield nearly identical results, rendering them equally useful in practical applications. An auxiliary consequence of Eqs.(24) and (25) is that, upon subtracting one from the other, the integral of 𝜕𝜕 𝑛𝑛 ( 𝐺𝐺𝜓𝜓 ) over the entire flat plane 𝑆𝑆 is found to vanish; that is, ∫ 𝜕𝜕 𝑛𝑛 [ 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) 𝜓𝜓 ( 𝒓𝒓 )]d 𝑠𝑠 𝑆𝑆1 = 0 . (26) We will have occasion to use this important identity in the following section.
6. Vector diffraction . Applying the Kirchhoff formula in Eq.(22), where the integral is over the closed surface 𝑆𝑆 = 𝑆𝑆 + 𝑆𝑆 , and the function 𝜓𝜓 ( 𝒓𝒓 ) is any scalar field that satisfies the Helmholtz equation, to a Cartesian component of the 𝐸𝐸 -field, say, 𝐸𝐸 𝑥𝑥 , we write 𝜋𝜋𝐸𝐸 𝑥𝑥 ( 𝒓𝒓 ) = ∫ [ 𝐸𝐸 𝑥𝑥 ( 𝒓𝒓 ) 𝜕𝜕 𝑛𝑛 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) − 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) 𝜕𝜕 𝑛𝑛 𝐸𝐸 𝑥𝑥 ( 𝒓𝒓 )]d 𝑠𝑠 𝑆𝑆 = ∫ [( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝐸𝐸 𝑥𝑥 − ( 𝒏𝒏� ∙ 𝜵𝜵𝐸𝐸 𝑥𝑥 ) 𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆 = ∫ [2( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝐸𝐸 𝑥𝑥 − 𝒏𝒏� ∙ 𝜵𝜵 ( 𝐺𝐺𝐸𝐸 𝑥𝑥 )]d 𝑠𝑠 𝑆𝑆 = 2 ∫ ( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝐸𝐸 𝑥𝑥 d 𝑠𝑠 𝑆𝑆 + ∫ 𝜵𝜵 ∙ 𝜵𝜵 ( 𝐺𝐺𝐸𝐸 𝑥𝑥 )d 𝒓𝒓 𝑉𝑉 . (27) Given that Eq.(27) is similarly satisfied by the remaining components 𝐸𝐸 𝑦𝑦 , 𝐸𝐸 𝑧𝑧 of the 𝐸𝐸 -field, the vectorial version of Kirchhoff’s formula may be written down straightforwardly. Algebraic manipulations (using standard vector calculus identities described in Appendix C) simplify the final result, yielding the following expression for the 𝐸𝐸 -field at the observation point: 𝑬𝑬 ( 𝒓𝒓 ) = (4 𝜋𝜋 ) −1 ∫ [( 𝒏𝒏� × 𝑬𝑬 ) × 𝜵𝜵𝐺𝐺 + ( 𝒏𝒏� ∙ 𝑬𝑬 ) 𝜵𝜵𝐺𝐺 + i 𝜔𝜔 ( 𝒏𝒏� × 𝑩𝑩 ) 𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆 . (28) Once again, it is easy to show that the contribution of the spherical (or hemi-spherical) surface 𝑆𝑆 to the overall integral in Eq.(28) is negligible. This is because, in the far field, 𝒏𝒏� ∙ 𝑬𝑬 → and 𝒏𝒏� × 𝑩𝑩 → 𝑬𝑬 𝑐𝑐⁄ , while | 𝑬𝑬 |~ 𝑒𝑒 i𝑘𝑘 𝑟𝑟 𝑟𝑟⁄ , 𝐺𝐺 ~ 𝑒𝑒 i𝑘𝑘 𝑟𝑟 𝑟𝑟⁄ and 𝜵𝜵𝐺𝐺 ~ − (i 𝑘𝑘 − 𝑟𝑟 −1 ) 𝑒𝑒 i𝑘𝑘 𝑟𝑟 𝒏𝒏� 𝑟𝑟⁄ . Consequently, 𝑬𝑬 ( 𝒓𝒓 ) = (4 𝜋𝜋 ) −1 ∫ [( 𝒏𝒏� × 𝑬𝑬 ) × 𝜵𝜵𝐺𝐺 + ( 𝒏𝒏� ∙ 𝑬𝑬 ) 𝜵𝜵𝐺𝐺 + i 𝜔𝜔 ( 𝒏𝒏� × 𝑩𝑩 ) 𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆1 . (29) A similar argument can be used to arrive at the vector Kirchhoff formula for the 𝐵𝐵 -field, namely, Gauss’ theorem; 𝒏𝒏� points into the volume 𝑉𝑉 𝜵𝜵 ( 𝜙𝜙𝜓𝜓 ) = 𝜙𝜙𝜵𝜵𝜓𝜓 + 𝜓𝜓𝜵𝜵𝜙𝜙 𝑩𝑩 ( 𝒓𝒓 ) = (4 𝜋𝜋 ) −1 ∫ [( 𝒏𝒏� × 𝑩𝑩 ) × 𝜵𝜵𝐺𝐺 + ( 𝒏𝒏� ∙ 𝑩𝑩 ) 𝜵𝜵𝐺𝐺 − i( 𝜔𝜔 𝑐𝑐 ⁄ )( 𝒏𝒏� × 𝑬𝑬 ) 𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆1 . (30) Needless to say, Eq.(30) could also be derived directly from Eq.(29), or vice versa, although the algebra becomes tedious at times; see Appendix D for one such derivation. We mention in passing that the arguments that led to Eqs.(29) and (30) could not be repeated for the vector potential 𝑨𝑨 ( 𝒓𝒓 ) , since in arriving at Eq.(28), in the step where 𝜵𝜵 ∙ 𝑬𝑬 or 𝜵𝜵 ∙ 𝑩𝑩 are set to zero (see Appendix C), we now have 𝜵𝜵 ∙ 𝑨𝑨 = i 𝜔𝜔𝜓𝜓 𝑐𝑐 ⁄ . ‡ In parallel with the arguments advanced previously in conjunction with the Rayleigh-Sommerfeld formulation for scalar fields, one may also modify Eqs.(29) and (30) by setting 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) = 0 and 𝜵𝜵𝐺𝐺 = 2 𝜕𝜕 𝑛𝑛 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) 𝒏𝒏� , provided that 𝑆𝑆 is a planar surface. This is equivalent to applying Eq.(25) directly to the 𝑥𝑥 , 𝑥𝑥 , 𝑧𝑧 components of the 𝐸𝐸 -field (or the 𝐵𝐵 -field). It is not permissible, however, to retain 𝐺𝐺 and remove 𝜵𝜵𝐺𝐺 (again, in the case of a planar 𝑆𝑆 ), because the gradient of 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 + ) + 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 − ) has a nonzero projection onto the 𝑥𝑥𝑥𝑥 -plane. In those special (yet important) cases where 𝑆𝑆 coincides with the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 , one could begin by applying either Eq.(24) or Eq.(25) to the 𝑥𝑥 , 𝑥𝑥 , 𝑧𝑧 components of the field under consideration. Manipulating the resulting equation with the aid of vector-algebraic identities in conjunction with the fact that 𝜵𝜵𝐺𝐺 = −𝜵𝜵 𝐺𝐺 , leads to vector diffraction formulas that could be useful under special circumstances. For instance, the vectorial equivalent of Eq.(25) yields 𝜋𝜋𝑬𝑬 ( 𝒓𝒓 ) = ∫ ( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝑬𝑬 ( 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 = ∫ [( 𝒏𝒏� × 𝑬𝑬 ) × 𝜵𝜵𝐺𝐺 + ( 𝑬𝑬 ∙ 𝜵𝜵𝐺𝐺 ) 𝒏𝒏� ]d 𝑠𝑠 𝑆𝑆1 = ∫ 𝜵𝜵 𝐺𝐺 × ( 𝒏𝒏� × 𝑬𝑬 )d 𝑠𝑠 𝑆𝑆1 + { ∫ [ 𝜵𝜵 ∙ ( 𝐺𝐺𝑬𝑬 ) − 𝐺𝐺𝜵𝜵 ∙ 𝑬𝑬 ]d 𝑠𝑠 𝑆𝑆1 } 𝒏𝒏� = ∫ 𝜵𝜵 × [( 𝒏𝒏� × 𝑬𝑬 ) 𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆1 + [ ∫ 𝜵𝜵 ∙ ( 𝐺𝐺𝑬𝑬 )d 𝑠𝑠 𝑆𝑆1 ] 𝒏𝒏� = 𝜵𝜵 × ∫ [ 𝒏𝒏� × 𝑬𝑬 ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 + �∬ [ 𝜕𝜕 𝑥𝑥 ( 𝐺𝐺𝐸𝐸 𝑥𝑥 ) + 𝜕𝜕 𝑦𝑦 ( 𝐺𝐺𝐸𝐸 𝑦𝑦 ) + 𝜕𝜕 𝑧𝑧 ( 𝐺𝐺𝐸𝐸 𝑧𝑧 )]d 𝑥𝑥 d 𝑥𝑥 ∞−∞ �𝒏𝒏� . (31) It is easy to see that the first two terms of the second integral on the right-hand side of Eq.(31) vanish since both 𝐺𝐺𝐸𝐸 𝑥𝑥 and 𝐺𝐺𝐸𝐸 𝑦𝑦 go to zero in the far out regions of the 𝑥𝑥𝑥𝑥 -plane. The vanishing of the third term, however, requires invoking Eq.(26) as applied to the 𝑧𝑧 -component of the 𝐸𝐸 -field. Note, as a matter of consistency, that on the left-hand side of Eq.(31), 𝜵𝜵 ∙ 𝑬𝑬 ( 𝒓𝒓 ) = 0 and that, on the right-hand side, the divergence of the curl is always zero. A similar argument can be advanced for the 𝐵𝐵 -field and, therefore, the following vector diffraction equations are generally valid for a planar surface 𝑆𝑆 : 𝑬𝑬 ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −1 𝜵𝜵 × ∫ [ 𝒏𝒏� × 𝑬𝑬 ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 . (32) 𝑩𝑩 ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −1 𝜵𝜵 × ∫ [ 𝒏𝒏� × 𝑩𝑩 ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 . (33) ‡ In setting 𝜵𝜵 ∙ 𝑬𝑬 = 0 , Maxwell’s first equation, 𝜀𝜀 𝜵𝜵 ∙ 𝑬𝑬 = 𝜌𝜌 total = 𝜌𝜌 free + 𝜌𝜌 bound , has been invoked, with the caveat that the surface 𝑆𝑆 is slightly detached from material bodies where electric charges of one kind or another may reside. No such caveat is needed, however, when setting 𝜵𝜵 ∙ 𝑩𝑩 = 0 , which is simply Maxwell’s fourth equation. The situation is quite different with the vector potential 𝑨𝑨 , since setting 𝜵𝜵 ∙ 𝑨𝑨 = 0 implies working in the Coulomb gauge. While the standard relations 𝑩𝑩 = 𝜵𝜵 × 𝑨𝑨 and 𝑬𝑬 = −𝜵𝜵𝜓𝜓 − 𝜕𝜕 𝑡𝑡 𝑨𝑨 remain valid in all gauges, the equations that relate 𝜓𝜓 and 𝑨𝑨 to the charge and current densities are gauge dependent. In particular, 𝑨𝑨 in the Coulomb gauge depends not only on the current-density 𝑱𝑱 total , but also on the charge-density 𝜌𝜌 total . While the charge-current continuity equation 𝜵𝜵 ∙ 𝑱𝑱 + 𝜕𝜕 𝑡𝑡 𝜌𝜌 = 0 can be used to arrive at a Helmholtz equation for 𝑨𝑨 ( 𝒓𝒓 , 𝑡𝑡 ) in the Coulomb gauge, the term appearing on the right-hand side of the equation will be the transverse current-density, which does not necessarily vanish in the free-space regions of the system under consideration. ( 𝒂𝒂 × 𝒃𝒃 ) × 𝒄𝒄 = ( 𝒂𝒂 ∙ 𝒄𝒄 ) 𝒃𝒃 − ( 𝒃𝒃 ∙ 𝒄𝒄 ) 𝒂𝒂
0 0 0 𝜵𝜵 ∙ ( 𝜓𝜓𝑽𝑽 ) = 𝜓𝜓𝜵𝜵 ∙ 𝑽𝑽 + 𝑽𝑽 ∙ 𝜵𝜵𝜓𝜓 𝜵𝜵 × ( 𝜓𝜓𝑽𝑽 ) = 𝜓𝜓𝜵𝜵 × 𝑽𝑽 + 𝜵𝜵𝜓𝜓 × 𝑽𝑽 In similar fashion, the vectorial equivalent of Eq.(24) yields 𝜋𝜋𝑬𝑬 ( 𝒓𝒓 ) = − ∫ 𝐺𝐺𝜕𝜕 𝑧𝑧 𝑬𝑬 d 𝑠𝑠 𝑆𝑆1 = − � 𝐺𝐺 [ 𝜕𝜕 𝑧𝑧 𝐸𝐸 𝑥𝑥 𝒙𝒙� + 𝜕𝜕 𝑧𝑧 𝐸𝐸 𝑦𝑦 𝒚𝒚� − ( 𝜕𝜕 𝑥𝑥 𝐸𝐸 𝑥𝑥 + 𝜕𝜕 𝑦𝑦 𝐸𝐸 𝑦𝑦 ) 𝒛𝒛� ]d 𝑠𝑠 𝑆𝑆1 = � [ 𝐸𝐸 𝑥𝑥 ( 𝜕𝜕 𝑧𝑧 𝐺𝐺𝒙𝒙� − 𝜕𝜕 𝑥𝑥 𝐺𝐺𝒛𝒛� ) + 𝐸𝐸 𝑦𝑦 ( 𝜕𝜕 𝑧𝑧 𝐺𝐺𝒚𝒚� − 𝜕𝜕 𝑦𝑦 𝐺𝐺𝒛𝒛� ) 𝑆𝑆1 −𝜕𝜕 𝑧𝑧 ( 𝐺𝐺𝐸𝐸 𝑥𝑥 ) 𝒙𝒙� − 𝜕𝜕 𝑧𝑧 ( 𝐺𝐺𝐸𝐸 𝑦𝑦 ) 𝒚𝒚� + 𝜕𝜕 𝑥𝑥 ( 𝐺𝐺𝐸𝐸 𝑥𝑥 ) 𝒛𝒛� + 𝜕𝜕 𝑦𝑦 ( 𝐺𝐺𝐸𝐸 𝑦𝑦 ) 𝒛𝒛� ]d 𝑠𝑠 = ∫ ( 𝒏𝒏� × 𝑬𝑬 ) × 𝜵𝜵𝐺𝐺 d 𝑠𝑠 𝑆𝑆1 − � �𝜕𝜕 𝑧𝑧 ( 𝐺𝐺𝐸𝐸 𝑥𝑥 ) 𝒙𝒙� + 𝜕𝜕 𝑧𝑧 ( 𝐺𝐺𝐸𝐸 𝑦𝑦 ) 𝒚𝒚� − 𝜕𝜕 𝑥𝑥 ( 𝐺𝐺𝐸𝐸 𝑥𝑥 ) 𝒛𝒛� − 𝜕𝜕 𝑦𝑦 ( 𝐺𝐺𝐸𝐸 𝑦𝑦 ) 𝒛𝒛�� d 𝑥𝑥 d 𝑥𝑥 ∞−∞ = ∫ 𝜵𝜵 𝐺𝐺 × ( 𝒏𝒏� × 𝑬𝑬 )d 𝑠𝑠 𝑆𝑆1 = 𝜵𝜵 × ∫ [ 𝒏𝒏� × 𝑬𝑬 ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 . (34) On the penultimate line of Eq.(34), the first two terms in the integral are seen to vanish when Eq.(26) is applied to the 𝑥𝑥 and 𝑥𝑥 components of the 𝐸𝐸 -field; the 3 rd and 4 th terms go to zero due to the vanishing of 𝐺𝐺𝐸𝐸 𝑥𝑥 and 𝐺𝐺𝐸𝐸 𝑦𝑦 in the far out regions of the 𝑥𝑥𝑥𝑥 -plane. Equation (34) thus yields the same expression for 𝑬𝑬 ( 𝒓𝒓 ) as the one reached via Eq.(31). Example 1 . A plane-wave 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = ( 𝐸𝐸 𝑥𝑥0 𝒙𝒙� + 𝐸𝐸 𝑦𝑦0 𝒚𝒚� + 𝐸𝐸 𝑧𝑧0 𝒛𝒛� ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦+𝑘𝑘 𝑧𝑧 𝑧𝑧−𝜔𝜔𝑡𝑡 ) arriving from the region 𝑧𝑧 < 0 is reflected from a perfectly conducting plane mirror located in the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 . The exact cancellation of the tangential components of the 𝐸𝐸 -field at the mirror surface means that the scattered 𝐸𝐸 -field in the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 + is given by § 𝑬𝑬 𝑠𝑠 ( 𝑥𝑥 , 𝑥𝑥 ) = − ( 𝐸𝐸 𝑥𝑥0 𝒙𝒙� + 𝐸𝐸 𝑦𝑦0 𝒚𝒚� + 𝐸𝐸 𝑧𝑧0 𝒛𝒛� ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) . (35) Beyond the mirror in the region 𝑧𝑧 > 0 , the scattered 𝐸𝐸 -field is obtained from Eq.(32), as follows: 𝜋𝜋𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) = 𝜵𝜵 × � 𝒏𝒏� × ( −𝐸𝐸 𝑥𝑥0 𝒙𝒙� − 𝐸𝐸 𝑦𝑦0 𝒚𝒚� − 𝐸𝐸 𝑧𝑧0 𝒛𝒛� ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) exp�i𝑘𝑘 [( 𝑥𝑥−𝑥𝑥 ) + ( 𝑦𝑦−𝑦𝑦 ) +𝑧𝑧 ] ½ � [( 𝑥𝑥−𝑥𝑥 ) + ( 𝑦𝑦−𝑦𝑦 ) +𝑧𝑧 ] ½ d 𝑥𝑥 d 𝑥𝑥 ∞−∞ = 𝜵𝜵 × ( 𝐸𝐸 𝑦𝑦0 𝒙𝒙� − 𝐸𝐸 𝑥𝑥0 𝒚𝒚� ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥 +𝑘𝑘 𝑦𝑦 𝑦𝑦 ) � 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) exp�i𝑘𝑘 ( 𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ) ½ � ( 𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ) ½ d 𝑥𝑥 d 𝑥𝑥 ∞−∞ . (36) The 2D Fourier transform appearing in Eq.(36) is readily found to be (see Appendix E): � exp�i𝑘𝑘 �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ��𝑥𝑥 +𝑦𝑦 +𝑧𝑧 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑥𝑥 d 𝑥𝑥 ∞−∞ = i(2 𝜋𝜋 𝑘𝑘 𝑧𝑧 ⁄ ) 𝑒𝑒 i𝑘𝑘 𝑧𝑧 𝑧𝑧 . (37) This is a valid equation whether the incident beam is of the propagating type (i.e., homogeneous plane-wave, with real-valued 𝑘𝑘 𝑧𝑧 ), or of the evanescent type (i.e., inhomogeneous, with imaginary 𝑘𝑘 𝑧𝑧 ). Substitution into Eq.(36) now yields 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) = 𝜵𝜵 × i( 𝐸𝐸 𝑦𝑦0 𝒙𝒙� − 𝐸𝐸 𝑥𝑥0 𝒚𝒚� ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥 +𝑘𝑘 𝑦𝑦 𝑦𝑦 +𝑘𝑘 𝑧𝑧 𝑧𝑧 ) 𝑘𝑘 𝑧𝑧 ⁄ = [ −𝐸𝐸 𝑥𝑥0 𝒙𝒙� − 𝐸𝐸 𝑦𝑦0 𝒚𝒚� + ( 𝑘𝑘 𝑥𝑥 𝐸𝐸 𝑥𝑥0 + 𝑘𝑘 𝑦𝑦0 𝐸𝐸 𝑦𝑦 ) 𝒛𝒛� 𝑘𝑘 𝑧𝑧 ⁄ ] 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥 +𝑘𝑘 𝑦𝑦 𝑦𝑦 +𝑘𝑘 𝑧𝑧 𝑧𝑧 ) = − ( 𝐸𝐸 𝑥𝑥0 𝒙𝒙� + 𝐸𝐸 𝑦𝑦0 𝒚𝒚� + 𝐸𝐸 𝑧𝑧0 𝒛𝒛� ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥 +𝑘𝑘 𝑦𝑦 𝑦𝑦 +𝑘𝑘 𝑧𝑧 𝑧𝑧 ) . (38) § Whereas on opposite facets of the mirror the scattered 𝐸𝐸 𝑥𝑥 (as well as the scattered 𝐸𝐸 𝑦𝑦 ) are identical, the presence of surface charges requires that the sign of the scattered 𝐸𝐸 𝑧𝑧 flip between 𝑧𝑧 = 0 − and 𝑧𝑧 = 0 + . The scattered 𝐸𝐸 -field amplitude is thus ( −𝐸𝐸 𝑥𝑥0 , −𝐸𝐸 𝑦𝑦0 , 𝐸𝐸 𝑧𝑧0 ) at 𝑧𝑧 = 0 − and ( −𝐸𝐸 𝑥𝑥0 , −𝐸𝐸 𝑦𝑦0 , −𝐸𝐸 𝑧𝑧0 ) at 𝑧𝑧 = 0 + . Similarly, the existence of surface currents requires the scattered 𝐵𝐵 -field amplitude to be ( 𝐵𝐵 𝑥𝑥0 , 𝐵𝐵 𝑦𝑦0 , −𝐵𝐵 𝑧𝑧0 ) at 𝑧𝑧 = 0 − and ( −𝐵𝐵 𝑥𝑥0 , −𝐵𝐵 𝑦𝑦0 , −𝐵𝐵 𝑧𝑧0 ) at 𝑧𝑧 = 0 + . 𝜵𝜵 ∙ 𝑬𝑬 = 0 → 𝜕𝜕 𝑧𝑧 𝐸𝐸 𝑧𝑧 = − ( 𝜕𝜕 𝑥𝑥 𝐸𝐸 𝑥𝑥 + 𝜕𝜕 𝑦𝑦 𝐸𝐸 𝑦𝑦 )
0 0 0 0
As expected, in the half-space 𝑧𝑧 > 0 , the scattered field precisely cancels out the incident field. This result is quite general and applies to any profile for the incident beam, not just plane-waves, for the simple reason that any incident beam can be expressed as a superposition of plane-waves. It should also be clear that, in the absence of the perfectly conducting reflector in the 𝑥𝑥𝑥𝑥 -plane, the distribution of the tangential 𝐸𝐸 -field (or the tangential 𝐵𝐵 -field) throughout the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 can be used to reconstruct the entire distribution in the 𝑧𝑧 > 0 half-space via either Eq.(32) or Eq.(33). Example 2 . Consider a screen in the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 consisting of obstructing segment(s) in the form of thin sheet(s) of perfect conductors and open regions otherwise. A monochromatic incident beam creates surface charges and surface currents on the metallic segment(s) of this planar screen. The scattered fields produced by the induced surface charges and currents can be described in terms of the scattered scalar and vector potentials 𝜓𝜓 𝑠𝑠 ( 𝒓𝒓 , 𝑡𝑡 ) and 𝑨𝑨 𝑠𝑠 ( 𝒓𝒓 , 𝑡𝑡 ) . Given that the induced surface current has no component along the 𝑧𝑧 -axis, the vector potential will likewise have only 𝑥𝑥 and 𝑥𝑥 components. Application of Eq.(24) to 𝐴𝐴 𝑠𝑠𝑥𝑥 ( 𝒓𝒓 ) and 𝐴𝐴 𝑠𝑠𝑦𝑦 ( 𝒓𝒓 ) yields 𝑨𝑨 𝑠𝑠 ( 𝒓𝒓 ) = − � exp ( i𝑘𝑘 | 𝒓𝒓−𝒓𝒓 |)| 𝒓𝒓−𝒓𝒓 | �𝜕𝜕 𝑧𝑧 𝐴𝐴 𝑠𝑠𝑥𝑥 𝒙𝒙� + 𝜕𝜕 𝑧𝑧 𝐴𝐴 𝑠𝑠𝑦𝑦 𝒚𝒚�� d 𝑠𝑠 𝑆𝑆1 . (39) Now, 𝑩𝑩 𝑠𝑠 ( 𝒓𝒓 ) = 𝜵𝜵 × 𝑨𝑨 𝑠𝑠 ( 𝒓𝒓 ) = −𝜕𝜕 𝑧𝑧 𝐴𝐴 𝑠𝑠𝑦𝑦 𝒙𝒙� + 𝜕𝜕 𝑧𝑧 𝐴𝐴 𝑠𝑠𝑥𝑥 𝒚𝒚� + ( 𝜕𝜕 𝑥𝑥 𝐴𝐴 𝑠𝑠𝑦𝑦 − 𝜕𝜕 𝑦𝑦 𝐴𝐴 𝑠𝑠𝑥𝑥 ) 𝒛𝒛� . Consequently, 𝑨𝑨 𝑠𝑠 ( 𝒓𝒓 ) = − � exp ( i𝑘𝑘 | 𝒓𝒓−𝒓𝒓 |)| 𝒓𝒓−𝒓𝒓 | ( 𝐵𝐵 𝑠𝑠𝑦𝑦 𝒙𝒙� − 𝐵𝐵 𝑠𝑠𝑥𝑥 𝒚𝒚� )d 𝑠𝑠 𝑆𝑆1 . (40) The symmetry of the scattered field ensures that 𝐵𝐵 𝑠𝑠𝑥𝑥 and 𝐵𝐵 𝑠𝑠𝑦𝑦 are zero in the open areas of the screen, so the integral in Eq.(40) need be evaluated only on the metallic surfaces of the screen. We will have 𝑩𝑩 𝑠𝑠 ( 𝒓𝒓 ) = 𝜵𝜵 × 𝑨𝑨 𝑠𝑠 ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −1 𝜵𝜵 × � [ 𝒏𝒏� × 𝑩𝑩 𝑠𝑠 ( 𝒓𝒓 )] exp ( i𝑘𝑘 | 𝒓𝒓−𝒓𝒓 |)| 𝒓𝒓−𝒓𝒓 | d 𝑠𝑠 metal . (41) In situations where a thin, flat metallic object acts as a scatterer, Eq.(41) provides a simple way to compute the scattered field provided, of course, that an estimate of the magnetic field at the metal surface (or, equivalently, an estimate of the induced surface current-density) can be obtained. Needless to say, considering that, in the absence of the scatterer, the continued propagation of the incident beam into the 𝑧𝑧 > 0 half-space can be reconstructed from the tangential component of the incident 𝐵𝐵 -field in the 𝑧𝑧 = 0 plane (see Example 1), one may add the incident 𝐵𝐵 -field to the scattered field of Eq.(41) and obtain, unsurprisingly, the general vector diffraction formula of Eq.(33). Example 3 . Figure 2 shows a perfectly conducting thin sheet residing in the upper half of the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 . The incident beam is a monochromatic plane-wave of frequency 𝜔𝜔 , wavelength 𝜆𝜆 = 2 𝜋𝜋𝑐𝑐 𝜔𝜔⁄ , linear-polarization aligned with 𝒚𝒚� , and propagation direction along the unit-vector 𝝈𝝈 inc = (sin 𝜃𝜃 inc ) 𝒙𝒙� + (cos 𝜃𝜃 inc ) 𝒛𝒛� , as follows: 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝐸𝐸 inc 𝒚𝒚� exp[i 𝑘𝑘 (sin 𝜃𝜃 inc 𝑥𝑥 + cos 𝜃𝜃 inc 𝑧𝑧 − 𝑐𝑐𝑡𝑡 )] . (42) Using Eq.(32), we derive the diffracted 𝐸𝐸 -field at the observation point 𝒓𝒓 = 𝑥𝑥 𝒙𝒙� + 𝑧𝑧 𝒛𝒛� . (The symmetry of the problem ensures that the field profile is independent of the 𝑥𝑥 -coordinate of the observation point.) In what follows, we use the 0 th and 1 st order Hankel functions of type 1, 0 𝐻𝐻 ( ) ( 𝜁𝜁 ) and 𝐻𝐻 ( ) ( 𝜁𝜁 ) , as well as the (complex) Fresnel integral 𝐹𝐹 ( 𝜁𝜁 ) = ∫ exp(i 𝑥𝑥 ) d 𝑥𝑥 ∞𝜁𝜁 . For a graphical representation of the Fresnel integral via the so-called Cornu spiral, see Appendix F. The following identities will be needed further below: � exp ( i�𝑥𝑥 +𝜁𝜁 ) �𝑥𝑥 +𝜁𝜁 d 𝑥𝑥 ∞−∞ = i 𝜋𝜋𝐻𝐻 ( ) ( 𝜁𝜁 ) . (43) dd𝜁𝜁 𝐻𝐻 ( ) ( 𝜁𝜁 ) = −𝐻𝐻 ( ) ( 𝜁𝜁 ) . (44) 𝐻𝐻 ( ) ( 𝜁𝜁 )~ � 𝑒𝑒 i ( 𝜁𝜁− ¾ 𝜋𝜋 ) , ( 𝜁𝜁 ≫ . (45) 𝐹𝐹 ( 𝜁𝜁 ) = ∫ exp(i 𝑥𝑥 ) d 𝑥𝑥 ∞0 − �∫ cos( 𝑥𝑥 ) d 𝑥𝑥 𝜁𝜁0 + i ∫ sin( 𝑥𝑥 ) d 𝑥𝑥 𝜁𝜁0 � = �𝜋𝜋 ⁄ 𝑒𝑒 i𝜋𝜋 4⁄ − �𝜋𝜋 ⁄ [ 𝐶𝐶 ( 𝜁𝜁 ) + i 𝑆𝑆 ( 𝜁𝜁 )] . (46) The 𝐸𝐸 -field at the observation point 𝒓𝒓 is found to be 𝑬𝑬 ( 𝒓𝒓 ) ≅ (2 𝜋𝜋 ) −1 𝜵𝜵 × � � ( 𝒛𝒛� × 𝐸𝐸 inc 𝒚𝒚� ) 𝑒𝑒 i𝑘𝑘 sin 𝜃𝜃 inc 𝑥𝑥 × exp�i𝑘𝑘 [( 𝑥𝑥−𝑥𝑥 ) +𝑦𝑦 +𝑧𝑧 ] ½ � [( 𝑥𝑥−𝑥𝑥 ) +𝑦𝑦 +𝑧𝑧 ] ½ d 𝑥𝑥 d 𝑥𝑥 ∞𝑦𝑦=−∞0𝑥𝑥=−∞ = − ½i 𝜵𝜵 × 𝐸𝐸 inc 𝒙𝒙� � 𝑒𝑒 i𝑘𝑘 sin 𝜃𝜃 inc 𝑥𝑥 𝐻𝐻 ( ) �𝑘𝑘 � ( 𝑥𝑥 − 𝑥𝑥 ) + 𝑧𝑧 � d 𝑥𝑥 = ½i 𝐸𝐸 inc 𝒚𝒚� � 𝑘𝑘 𝑧𝑧 [( 𝑥𝑥−𝑥𝑥 ) +𝑧𝑧 ] ½ 𝑒𝑒 i𝑘𝑘 sin 𝜃𝜃 inc 𝑥𝑥 𝐻𝐻 ( ) �𝑘𝑘 � ( 𝑥𝑥 − 𝑥𝑥 ) + 𝑧𝑧 � d 𝑥𝑥 ≅ i𝑒𝑒 −i3𝜋𝜋 4⁄ 𝐸𝐸 inc 𝒚𝒚�√2𝜋𝜋 � �𝑘𝑘 𝑧𝑧 [( 𝑥𝑥−𝑥𝑥 ) +𝑧𝑧 ] exp(i 𝑘𝑘 sin 𝜃𝜃 inc 𝑥𝑥 ) exp � i 𝑘𝑘 � ( 𝑥𝑥 − 𝑥𝑥 ) + 𝑧𝑧 � d 𝑥𝑥 . (47) Fig.2 . A perfectly conducting thin sheet sits in the upper half of the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 . The incident plane-wave, which is linearly polarized along the 𝑥𝑥 -axis, has amplitude 𝐸𝐸 inc , frequency 𝜔𝜔 , wave-number 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ , and propagation direction 𝝈𝝈 inc = sin 𝜃𝜃 inc 𝒙𝒙� + cos 𝜃𝜃 inc 𝒛𝒛� . The observation point 𝒓𝒓 is in the 𝑥𝑥𝑧𝑧 -plane. 𝜃𝜃 inc 𝑥𝑥 𝑧𝑧 𝑥𝑥 ∙ × 𝒓𝒓 = 𝑥𝑥 𝒙𝒙� + 𝑧𝑧 𝒛𝒛� 𝐸𝐸 inc 𝒚𝒚� ∙ G&R -11 G&R -6 G&R -3 G&R -2,3 𝐻𝐻 ( ) ( 𝜁𝜁 ) given by Eq.(45). Assuming that 𝑧𝑧 ≫ | 𝑥𝑥 − 𝑥𝑥 | , we proceed by invoking the following approximation: exp � i 𝑘𝑘 � ( 𝑥𝑥 − 𝑥𝑥 ) + 𝑧𝑧 � [( 𝑥𝑥 − 𝑥𝑥 ) + 𝑧𝑧 ] � ≅ 𝑧𝑧 𝑒𝑒 i𝑘𝑘 [ 𝑧𝑧 + ( 𝑥𝑥−𝑥𝑥 ) ⁄ ] . (48) Note that this approximation is not accurate when | 𝑥𝑥 − 𝑥𝑥 | acquires large values; however, the rapid phase variations of the integrand in Eq.(47) ensure that the contributions to the integral at points 𝑥𝑥 that are far from 𝑥𝑥 are insignificant. Substitution from Eq.(48) into Eq.(47) yields 𝑬𝑬 ( 𝒓𝒓 ) ≅ �𝑘𝑘 𝐸𝐸 inc 𝒚𝒚��2𝜋𝜋𝑧𝑧 𝑒𝑒 i ( 𝑘𝑘 𝑧𝑧 −𝜋𝜋 4⁄ ) ∫ exp(i 𝑘𝑘 sin 𝜃𝜃 inc 𝑥𝑥 ) exp[i 𝑘𝑘 ( 𝑥𝑥 − 𝑥𝑥 ) (2 𝑧𝑧 ) ⁄ ] d 𝑥𝑥 = �𝑘𝑘 𝐸𝐸 inc 𝒚𝒚��2𝜋𝜋𝑧𝑧 𝑒𝑒 i ( 𝑘𝑘 𝑧𝑧 −𝜋𝜋 4⁄ ) exp[i 𝑘𝑘 ( 𝑥𝑥 sin 𝜃𝜃 inc − ½ 𝑧𝑧 sin 𝜃𝜃 inc )] × ∫ exp[i 𝑘𝑘 ( 𝑥𝑥 − 𝑥𝑥 + 𝑧𝑧 sin 𝜃𝜃 inc ) (2 𝑧𝑧 ) ⁄ ] d 𝑥𝑥 = 𝐸𝐸 inc 𝒚𝒚�√𝜋𝜋𝑒𝑒 i𝜋𝜋 4⁄ 𝑒𝑒 i𝑘𝑘 ( 𝑥𝑥 sin 𝜃𝜃 inc +𝑧𝑧 − ½ 𝑧𝑧 sin 𝜃𝜃 inc ) ∫ exp(i 𝑥𝑥 ) d 𝑥𝑥 ∞ �𝑘𝑘 ⁄ ( 𝑥𝑥 −𝑧𝑧 sin 𝜃𝜃 inc ) = 𝐹𝐹��𝜋𝜋 ( 𝜆𝜆 𝑧𝑧 ) ⁄ ( 𝑥𝑥 − 𝑧𝑧 sin 𝜃𝜃 inc ) �√𝜋𝜋 exp ( i𝜋𝜋 4⁄ ) 𝐸𝐸 inc 𝒚𝒚� 𝑒𝑒 i𝑘𝑘 [ 𝑥𝑥 sin 𝜃𝜃 inc +𝑧𝑧 ( ½ sin 𝜃𝜃 inc )] . (49) At the edge of the geometric shadow, i.e., the straight line where 𝑥𝑥 = 𝑧𝑧 sin 𝜃𝜃 inc , we have 𝐹𝐹 (0) = ½ √𝜋𝜋𝑒𝑒 i𝜋𝜋 4⁄ . On this line, the 𝐸𝐸 -field amplitude is ½ 𝐸𝐸 inc . Above the shadow’s edge, the field amplitude steadily declines, whereas below the edge, there occur a large number of oscillations before the field settles into what is essentially the incident plane-wave. Figure 3 shows a typical plot of the far-field intensity (i.e., the square of the 𝐸𝐸 -field amplitude) versus the distance (along the 𝑥𝑥 -axis at a fixed value of 𝑧𝑧 ) from the edge of the geometric shadow. Fig.3 . Normalized intensity in the far field of a sharp edge as a function of distance along the 𝑥𝑥 -axis (at fixed 𝑧𝑧 ) from the edge of the geometric shadow, where 𝑥𝑥 = 𝑧𝑧 sin 𝜃𝜃 inc . Example 4 . Shown in Fig.4 is a circular aperture of radius 𝑅𝑅 within an otherwise opaque screen located in the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 . A plane-wave 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑬𝑬 inc exp[i( 𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )] , where the unit-vector 𝝈𝝈 inc specifies the direction of incidence, arrives at the aperture from the left-hand side. Maxwell’s 3 rd equation identifies the incident 𝐵𝐵 -field as 𝑩𝑩 = 𝝈𝝈 inc × 𝑬𝑬 𝑐𝑐⁄ . The observation point 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 is sufficiently far from the aperture for the following approximation to apply: − − − −
5 0 5 10
Distance from the edge of geometric shadow (arbitrary units) N o r m a li ze d i n t e n s it y 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) = exp ( i𝑘𝑘 | 𝒓𝒓−𝒓𝒓 |)| 𝒓𝒓−𝒓𝒓 | = exp � i 𝑘𝑘 �𝑟𝑟 + 𝑟𝑟 − 𝒓𝒓 ∙ 𝒓𝒓 � | 𝒓𝒓 − 𝒓𝒓 | � ≅ exp [ i𝑘𝑘 ( 𝑟𝑟 − 𝝈𝝈 ∙ 𝒓𝒓 )] 𝑟𝑟 . (50) Let us assume that the screen is a thin sheet of a perfect conductor on whose surface the tangential 𝐸𝐸 -field necessarily vanishes. The appropriate diffraction equation will then be Eq.(32), with the tangential 𝐸𝐸 -field outside the aperture allowed to vanish (i.e., 𝒏𝒏� × 𝑬𝑬 = 0 on the opaque parts of the screen). Approximating the 𝐸𝐸 -field within the aperture with that of the incident plane-wave, we will have ∫ [ 𝒏𝒏� × 𝑬𝑬 ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 ≅ ( 𝒛𝒛� × 𝑬𝑬 inc ) � exp(i 𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 ) × exp [ i𝑘𝑘 ( 𝑟𝑟 − 𝝈𝝈 ∙ 𝒓𝒓 )] 𝑟𝑟 d 𝑠𝑠 aperture = exp ( i𝑘𝑘 𝑟𝑟 ) 𝑟𝑟 ( 𝒛𝒛� × 𝑬𝑬 inc ) ∫ exp[i 𝑘𝑘 ( 𝝈𝝈 inc − 𝝈𝝈 ) ∙ 𝒓𝒓 ] d 𝑠𝑠 aperture = exp ( i𝑘𝑘 𝑟𝑟 ) 𝑟𝑟 ( 𝒛𝒛� × 𝑬𝑬 inc ) ∫ ∫ exp(i 𝑘𝑘 𝜁𝜁𝑟𝑟 cos 𝜑𝜑 ) 𝑟𝑟 d 𝜑𝜑 d 𝑟𝑟 . (51) Fig.4 . A circular aperture of radius 𝑅𝑅 inside an otherwise opaque screen located at 𝑧𝑧 = 0 is illuminated by a plane-wave whose 𝐸𝐸 -field amplitude and propagation direction are specified as 𝑬𝑬 inc and 𝝈𝝈 inc , respectively. The observation point 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 is in the far field; that is, 𝑟𝑟 ≫ 𝑅𝑅 . The unit-vectors 𝝈𝝈 inc and 𝝈𝝈 have respective polar coordinates ( 𝜃𝜃 inc , 𝜑𝜑 inc ) and ( 𝜃𝜃 , 𝜑𝜑 ) . The projection of the vector 𝝈𝝈 inc − 𝝈𝝈 onto the 𝑥𝑥𝑥𝑥 -plane of the aperture is a vector of length 𝜁𝜁 that makes an angle 𝜑𝜑 with the position vector 𝒓𝒓 = 𝑥𝑥𝒙𝒙� + 𝑥𝑥𝒚𝒚� . Here, 𝜁𝜁 is the magnitude of the projection of 𝝈𝝈 inc − 𝝈𝝈 onto the 𝑥𝑥𝑥𝑥 -plane of the aperture, while 𝜑𝜑 is the angle between that projection and the position vector 𝒓𝒓 = 𝑥𝑥𝒙𝒙� + 𝑥𝑥𝒚𝒚� ; that is, 𝝈𝝈 inc − 𝝈𝝈 = (sin 𝜃𝜃 inc cos 𝜑𝜑 inc − sin 𝜃𝜃 cos 𝜑𝜑 ) 𝒙𝒙� + (sin 𝜃𝜃 inc sin 𝜑𝜑 inc − sin 𝜃𝜃 sin 𝜑𝜑 ) 𝒚𝒚� + (cos 𝜃𝜃 inc − cos 𝜃𝜃 ) 𝒛𝒛� . (52) 𝜁𝜁 = � sin 𝜃𝜃 inc + sin 𝜃𝜃 − 𝜃𝜃 inc sin 𝜃𝜃 cos( 𝜑𝜑 inc − 𝜑𝜑 ) . (53) Consequently, 𝑧𝑧 𝑥𝑥 𝑥𝑥 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 𝒓𝒓 𝒌𝒌 inc = 𝑘𝑘 𝝈𝝈 inc 𝑬𝑬 inc 𝒏𝒏� ∫ [ 𝒏𝒏� × 𝑬𝑬 ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 ≅ ( i𝑘𝑘 𝑟𝑟 ) 𝑟𝑟 ( 𝒛𝒛� × 𝑬𝑬 inc ) ∫ 𝑟𝑟𝐽𝐽 ( 𝑘𝑘 𝜁𝜁𝑟𝑟 )d 𝑟𝑟 𝑅𝑅0 = ( i𝑘𝑘 𝑟𝑟 ) 𝑘𝑘 𝑟𝑟 𝜁𝜁 𝐽𝐽 ( 𝑘𝑘 𝜁𝜁𝑅𝑅 ) 𝒛𝒛� × 𝑬𝑬 inc . (54) The 𝐸𝐸 -field at the observation point 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 (which is in the far field of the aperture, i.e., 𝑟𝑟 ≫ 𝑅𝑅 ) is now found by computing the curl of the expression on the right-hand side of Eq.(54). Considering that 𝒛𝒛� × 𝑬𝑬 inc is independent of 𝒓𝒓 , we use the vector identity 𝜵𝜵 × ( 𝜓𝜓𝑽𝑽 ) = 𝜵𝜵𝜓𝜓 × 𝑽𝑽 , then ignore the small (far field) contributions to 𝜵𝜵𝜓𝜓 due to the dependence of 𝜁𝜁 on ( 𝜃𝜃 , 𝜑𝜑 ) , to arrive at 𝑬𝑬 ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −1 𝜵𝜵 × ∫ [ 𝒏𝒏� × 𝑬𝑬 ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆1 ≅ ( 𝑅𝑅 𝑘𝑘 𝜁𝜁⁄ ) 𝐽𝐽 ( 𝑘𝑘 𝜁𝜁𝑅𝑅 ) 𝜵𝜵 [exp(i 𝑘𝑘 𝑟𝑟 ) 𝑟𝑟 ⁄ ] × ( 𝒛𝒛� × 𝑬𝑬 inc ) = ( 𝑅𝑅 𝑘𝑘 𝜁𝜁⁄ ) 𝐽𝐽 ( 𝑘𝑘 𝜁𝜁𝑅𝑅 )(i 𝑘𝑘 − 𝑟𝑟 −1 )[exp(i 𝑘𝑘 𝑟𝑟 ) 𝑟𝑟 ⁄ ] 𝝈𝝈 × ( 𝒛𝒛� × 𝑬𝑬 inc ) ≅ i𝜋𝜋𝐽𝐽 ( 𝑘𝑘 𝜁𝜁𝜋𝜋 ) 𝑒𝑒 i𝑘𝑘0𝑟𝑟0 𝑟𝑟 𝜁𝜁 [( 𝑬𝑬 inc ∙ 𝝈𝝈 ) 𝒛𝒛� − cos( 𝜃𝜃 ) 𝑬𝑬 inc ] . (55) The approximate nature of these calculations should be borne in mind when comparing the various estimates of an observed field obtained via different routes. For instance, had we started with Eq.(33) and proceeded by setting 𝒏𝒏� × 𝑩𝑩 = 0 on the opaque areas of the screen, we would have arrived at the following estimate of the observed 𝐸𝐸 -field: 𝑬𝑬 ( 𝒓𝒓 ) ≅ i𝜋𝜋𝐽𝐽 ( 𝑘𝑘 𝜁𝜁𝜋𝜋 ) 𝑒𝑒 i𝑘𝑘0𝑟𝑟0 𝑟𝑟 𝜁𝜁 {( 𝑬𝑬 inc ∙ 𝒛𝒛� )[ 𝝈𝝈 inc − ( 𝝈𝝈 inc ∙ 𝝈𝝈 ) 𝝈𝝈 ] − cos 𝜃𝜃 inc [ 𝑬𝑬 inc − ( 𝑬𝑬 inc ∙ 𝝈𝝈 ) 𝝈𝝈 ]} . (56) The differences between Eqs.(55) and (56), which, in general, are not insignificant, can be traced to the assumptions regarding the nature of the opaque screen and the approximations involved in equating the 𝑬𝑬 and 𝑩𝑩 fields within the aperture to those of the incident plane-wave.
7. Sommerfeld’s analysis of diffraction from a perfectly conducting half-plane . A rare example of an exact solution of Maxwell’s equations as applied to EM diffraction was published by A. Sommerfeld in 1896. The simplified version of Sommerfeld’s original analysis presented in this section closely parallels that of Ref. [1], Chapter 11. Consider an EM plane-wave propagating in free space and illuminating a thin, perfectly conducting screen that sits in the upper half of the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 . The geometry of the system is shown in Fig.5, where the incident 𝑘𝑘 -vector is denoted by 𝒌𝒌 inc , the oscillation frequency of the monochromatic wave is 𝜔𝜔 , the speed of light in vacuum is 𝑐𝑐 , the wave-number is 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ , and the unit-vector along the direction of incidence is 𝝈𝝈 inc . The plane-wave is linearly polarized with its 𝐸𝐸 -field amplitude 𝐸𝐸 along the 𝑥𝑥 -axis and 𝐻𝐻 -field amplitude 𝐻𝐻 = 𝐸𝐸 / 𝑍𝑍 in the 𝑥𝑥𝑧𝑧 -plane of incidence. ( 𝑍𝑍 = �𝜇𝜇 𝜀𝜀 ⁄ is the impedance of free space.) The electric current density induced in the semi-infinite screen is denoted by 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) . In the chosen geometry, all the fields (incident as well as scattered) are uniform along the 𝑥𝑥 -axis; consequently, it suffices to specify the position of an arbitrary point in the Cartesian 𝑥𝑥𝑥𝑥𝑧𝑧 space by its 𝑥𝑥 and 𝑧𝑧 coordinates only; that is, 𝒓𝒓 = 𝑥𝑥𝒙𝒙� + 𝑧𝑧𝒛𝒛� = 𝑟𝑟 (cos 𝜃𝜃 𝒙𝒙� + sin 𝜃𝜃 𝒛𝒛� ) . (57) Here, 𝑟𝑟 = √𝑥𝑥 + 𝑧𝑧 , and the angle 𝜃𝜃 is measured clockwise from the positive 𝑥𝑥 -axis. The range of 𝜃𝜃 is (0, 𝜋𝜋 ) for points 𝒓𝒓 on the right-hand side of the screen, and ( −𝜋𝜋 , 0) for the points 𝐽𝐽 ( ∙ ) and 𝐽𝐽 ( ∙ ) are Bessel functions of the first kind, orders 0 and 1. 𝒌𝒌 inc makes an angle 𝜃𝜃 ∈ (0, 𝜋𝜋 ) with the positive 𝑥𝑥 -axis. The incident plane-wave is, thus, fully specified by the following equations: 𝒌𝒌 inc = 𝑘𝑘 𝑥𝑥 𝒙𝒙� + 𝑘𝑘 𝑧𝑧 𝒛𝒛� = 𝑘𝑘 𝝈𝝈 inc = 𝑘𝑘 (cos 𝜃𝜃 𝒙𝒙� + sin 𝜃𝜃 𝒛𝒛� ) . (58) 𝑬𝑬 inc ( 𝒓𝒓 , 𝑡𝑡 ) = 𝐸𝐸 𝒚𝒚� exp[i( 𝒌𝒌 inc ∙ 𝒓𝒓 − 𝜔𝜔𝑡𝑡 )] = 𝐸𝐸 𝒚𝒚� exp[i 𝑘𝑘 ( 𝝈𝝈 inc ∙ 𝒓𝒓 − 𝑐𝑐𝑡𝑡 )] = 𝐸𝐸 𝒚𝒚�𝑒𝑒 i𝑘𝑘 𝑟𝑟 cos ( 𝜃𝜃−𝜃𝜃 ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 . (59) 𝑯𝑯 inc ( 𝒓𝒓 , 𝑡𝑡 ) = − ( 𝐸𝐸 𝑍𝑍 ⁄ )(sin 𝜃𝜃 𝒙𝒙� − cos 𝜃𝜃 𝒛𝒛� ) 𝑒𝑒 i𝑘𝑘 𝑟𝑟 cos ( 𝜃𝜃−𝜃𝜃 ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 . (60) Fig.5 . A plane, monochromatic EM wave propagating along the unit-vector 𝝈𝝈 inc = cos 𝜃𝜃 𝒙𝒙� + sin 𝜃𝜃 𝒛𝒛� arrives at a thin, semi-infinite, perfectly electrically conducting screen located in the upper half of the 𝑥𝑥𝑥𝑥 -plane. The plane-wave is linearly polarized, with its 𝐸𝐸 -field aligned with the 𝑥𝑥 -axis, while its 𝐻𝐻 -field has components along the 𝑥𝑥 and 𝑧𝑧 directions. The induced current sheet, denoted by 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) , oscillates parallel to the incident 𝐸𝐸 -field along the direction of the 𝑥𝑥 -axis. (The system depicted here is essentially the same as that in Fig.2, with the exception of the angle of incidence 𝜃𝜃 being the complement of 𝜃𝜃 inc of Fig.2.) With the aid of the step-function step( 𝑥𝑥 ) and Dirac’s delta-function 𝛿𝛿 ( 𝑧𝑧 ) , we now express the electric current density 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) excited on the thin-sheet conductor as a one-dimensional Fourier transform, namely, 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) = step( 𝑥𝑥 ) 𝐽𝐽 𝑠𝑠 ( 𝑥𝑥 ) 𝛿𝛿 ( 𝑧𝑧 ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 𝒚𝒚� = �∫ 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) 𝑒𝑒 i𝑘𝑘 𝜎𝜎 𝑥𝑥 𝑥𝑥 d 𝜎𝜎 𝑥𝑥 ∞−∞ �𝛿𝛿 ( 𝑧𝑧 ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 𝒚𝒚� . (61) In the above equation, 𝜎𝜎 𝑥𝑥 represents spatial frequency along the 𝑥𝑥 -axis, and 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) is the (complex) amplitude of the induced surface-current-density along 𝒚𝒚� , having spatial and temporal frequencies 𝜎𝜎 𝑥𝑥 and 𝜔𝜔 , respectively. The first goal of our analysis is to find the function 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) such that its Fourier transform vanishes along the negative 𝑥𝑥 -axis — as demanded by the step-function in Eq.(61) — while its radiated 𝐸𝐸 -field cancels the incident plane-wave’s 𝐸𝐸 -field at the surface of the screen along the positive 𝑥𝑥 -axis. Now, a current sheet 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) 𝑒𝑒 i ( 𝑘𝑘 𝜎𝜎 𝑥𝑥 𝑥𝑥−𝜔𝜔𝑡𝑡 ) 𝛿𝛿 ( 𝑧𝑧 ) 𝒚𝒚� that fills the entire 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 radiates EM fields into the surrounding free space that can easily be shown to have the following structure: 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = − ½( 𝑍𝑍 𝑘𝑘 𝒚𝒚� 𝑘𝑘 𝑧𝑧 ⁄ ) 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) exp[i( 𝑘𝑘 𝑥𝑥 𝑥𝑥 + 𝑘𝑘 𝑧𝑧 | 𝑧𝑧 | − 𝜔𝜔𝑡𝑡 )] . (62) 𝑯𝑯 ( 𝒓𝒓 , 𝑡𝑡 ) = ½ 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 )[± 𝒙𝒙� − ( 𝑘𝑘 𝑥𝑥 𝑘𝑘 𝑧𝑧 ⁄ ) 𝒛𝒛� ] exp[i( 𝑘𝑘 𝑥𝑥 𝑥𝑥 + 𝑘𝑘 𝑧𝑧 | 𝑧𝑧 | − 𝜔𝜔𝑡𝑡 )] . (63) 𝑧𝑧 𝑥𝑥 𝑯𝑯 𝑬𝑬 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) 𝜃𝜃 𝒌𝒌 inc = 𝑘𝑘 𝝈𝝈 inc ∙ 𝑥𝑥 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ is the wave-number in free space, while 𝑘𝑘 𝑥𝑥 = 𝑘𝑘 𝜎𝜎 𝑥𝑥 , and 𝑘𝑘 𝑧𝑧 = �𝑘𝑘 − 𝑘𝑘 𝑥𝑥2 are the 𝑘𝑘 -vector components along the 𝑥𝑥 and 𝑧𝑧 axes. In general, 𝑘𝑘 𝑧𝑧 must be real and positive when | 𝑘𝑘 𝑥𝑥 | ≤ 𝑘𝑘 , and imaginary and positive otherwise. The ± signs associated with 𝐻𝐻 𝑥𝑥 indicate that the plus sign must be used for the half-space on the right-hand side of the sheet, where 𝑧𝑧 > 0 , while the minus sign is reserved for the left-hand side, where 𝑧𝑧 < 0 . The discontinuity of 𝐻𝐻 𝑥𝑥 at 𝑧𝑧 = 0 thus equals the surface-current-density 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) along the 𝑥𝑥 -direction, in compliance with Maxwell’s requisite boundary conditions. Needless to say, Eqs.(62) and (63) represent a single EM plane-wave on either side of the screen’s 𝑥𝑥𝑥𝑥 -plane, which satisfy the symmetry requirement of radiation from the current sheet, and with the tangential component 𝐻𝐻 𝑥𝑥 of the radiated 𝐻𝐻 -field chosen to satisfy the requisite boundary condition at the plane of the surface current. The plane-waves emanating to the right and left of the 𝑥𝑥𝑥𝑥 -plane of the current sheet are homogeneous when 𝑘𝑘 𝑧𝑧 is real, and inhomogeneous (or evanescent) when 𝑘𝑘 𝑧𝑧 is imaginary. When working in the complex 𝜎𝜎 𝑥𝑥 -plane, we must ensure that 𝑘𝑘 𝑧𝑧 has the correct sign for all values of the real parameter 𝑘𝑘 𝑥𝑥 = 𝑘𝑘 𝜎𝜎 𝑥𝑥 from −∞ to ∞ . Considering that 𝑘𝑘 𝑧𝑧 𝑘𝑘 ⁄ is the square root of the product of (1 − 𝜎𝜎 𝑥𝑥 ) and (1 + 𝜎𝜎 𝑥𝑥 ) , we choose for both of these complex numbers the range of phase angles ( −𝜋𝜋 , 𝜋𝜋 ] , as depicted in Fig.6(a). The corresponding branch-cuts thus appear as the semi-infinite line segments ( −∞ , − and (1, ∞ ) , and the integration path along the real axis within the 𝜎𝜎 𝑥𝑥 -plane, shown in Fig.6(b), will consist of semi-infinite line segments slightly above and slightly below the real axis, as well as a short segment of the real-axis connecting − to . This choice of the integration path ensures that 𝑘𝑘 𝑧𝑧 = 𝑘𝑘 � − 𝜎𝜎 𝑥𝑥2 acquires the correct sign for all values of 𝜎𝜎 𝑥𝑥 . Fig.6 . (a) In the complex 𝜎𝜎 𝑥𝑥 -plane, where the phase of the complex numbers 𝜎𝜎 𝑥𝑥 is measured counterclock-wise from the positive 𝜎𝜎 𝑥𝑥′ axis, the range of both angles is confined to the ( −𝜋𝜋 , 𝜋𝜋 ] interval. (b) The integration path along the 𝜎𝜎 𝑥𝑥′ axis is adjusted by shifting the part from −∞ to − slightly upward, and the part from to ∞ slightly downward, so that 𝑘𝑘 𝑧𝑧 𝑘𝑘 ⁄ = � (1 − 𝜎𝜎 𝑥𝑥 )(1 + 𝜎𝜎 𝑥𝑥 ) has the correct sign everywhere. The pole at 𝜎𝜎 𝑥𝑥 =cos 𝜃𝜃 is handled by locally deforming the contour into a semi-circular path below the real axis. For reasons that will become clear as we proceed, Sommerfeld suggested the following mathematical form for the surface current-density 𝒥𝒥 as a function of 𝜎𝜎 𝑥𝑥 : 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) = 𝒥𝒥 � − 𝜎𝜎 𝑥𝑥 ( 𝜎𝜎 𝑥𝑥 − cos 𝜃𝜃 ) � . (64) The domain of this function is the slightly deformed real axis of the 𝜎𝜎 𝑥𝑥 -plane depicted in Fig.6(b). The proposed function has a simple pole at 𝜎𝜎 𝑥𝑥 = cos 𝜃𝜃 , where 𝜃𝜃 is the orientation angle of 𝒌𝒌 inc shown in Fig.5, and a (complex) constant coefficient 𝒥𝒥 that will be determined 𝜎𝜎 𝑥𝑥′ − 𝜎𝜎 𝑥𝑥″ 𝜎𝜎 𝑥𝑥 𝜎𝜎 𝑥𝑥 − 𝜎𝜎 𝑥𝑥 𝜎𝜎 𝑥𝑥′ − 𝜎𝜎 𝑥𝑥″ cos 𝜃𝜃 𝜀𝜀 𝜀𝜀 (a) (b) 6 shortly. In addition, 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) contains the term � − 𝜎𝜎 𝑥𝑥 , whose branch-cut in the system of Fig.6(a) is the semi-infinite line-segment (1, ∞ ) along the real axis of the 𝜎𝜎 𝑥𝑥 -plane. It is now possible to demonstrate that the proposed 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) satisfies its first required property, namely, ∫ 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) 𝑒𝑒 i𝑘𝑘 𝜎𝜎 𝑥𝑥 𝑥𝑥 d 𝜎𝜎 𝑥𝑥 ∞−∞ = 𝒥𝒥 � �1−𝜎𝜎 𝑥𝑥 𝜎𝜎 𝑥𝑥 − cos 𝜃𝜃 𝑒𝑒 i𝑘𝑘 𝜎𝜎 𝑥𝑥 𝑥𝑥 d 𝜎𝜎 𝑥𝑥 ∞−∞ = 0, ( 𝑥𝑥 < 0) . (65) For 𝑥𝑥 < 0 , the integration contour of Fig.6(b) can be closed with a large semi-circular path in the lower half of the 𝜎𝜎 𝑥𝑥 -plane. The only part of the integrand in Eq.(65) that requires a branch-cut is � − 𝜎𝜎 𝑥𝑥 , whose branch-cut is the line segment (1, ∞ ) . The integration contour of Fig.6(b), when closed in the lower-half of the 𝜎𝜎 𝑥𝑥 -plane, does not contain this branch-cut. Moreover, the pole at 𝜎𝜎 𝑥𝑥 = cos 𝜃𝜃 is outside the closed loop of integration, and the contributions to the integral by the singular points 𝜎𝜎 𝑥𝑥 = ±1 are zero. Consequently, the current-density 𝑱𝑱 ( 𝒓𝒓 , 𝑡𝑡 ) in the lower half of the 𝑥𝑥𝑥𝑥 -plane turns out to be zero, exactly as required. The radiated 𝐸𝐸 -field, a superposition of contributions from the various 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) in accordance with Eqs.(62) and (64), must cancel out the incident 𝐸𝐸 -field at the surface of the screen; that is, 𝐸𝐸 𝑦𝑦 ( 𝑥𝑥 , 𝑥𝑥 , 𝑧𝑧 = 0) = � 𝐸𝐸 𝑦𝑦 ( 𝜎𝜎 𝑥𝑥 ) 𝑒𝑒 i𝑘𝑘 𝜎𝜎 𝑥𝑥 𝑥𝑥 d 𝜎𝜎 𝑥𝑥 ∞−∞ = − ½ 𝑍𝑍 𝑘𝑘 ∫ [ 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) 𝑘𝑘 𝑧𝑧 ⁄ ] 𝑒𝑒 i𝑘𝑘 𝜎𝜎 𝑥𝑥 𝑥𝑥 d 𝜎𝜎 𝑥𝑥 ∞−∞ = − ½ 𝑍𝑍 𝒥𝒥 � 𝑒𝑒 i𝑘𝑘0𝜎𝜎𝑥𝑥𝑥𝑥 �1+𝜎𝜎 𝑥𝑥 ( 𝜎𝜎 𝑥𝑥 − cos 𝜃𝜃 ) d 𝜎𝜎 𝑥𝑥 ∞−∞ = −𝐸𝐸 𝑒𝑒 i𝑘𝑘 cos 𝜃𝜃 𝑥𝑥 , ( 𝑥𝑥 > 0) . (66) For 𝑥𝑥 > 0 , the contour of integration in Eq.(66) can be closed with a large semi-circle in the upper-half of the 𝜎𝜎 𝑥𝑥 -plane. The branch-cut for � 𝜎𝜎 𝑥𝑥 in the denominator of the integrand is the line-segment ( −∞ , − , which is below the integration path and, therefore, irrelevant for a contour that closes in the upper-half-plane. For the singularities at 𝜎𝜎 𝑥𝑥 = ±1 , the residues are zero, whereas for the pole at 𝜎𝜎 𝑥𝑥 = cos 𝜃𝜃 , the residue is 𝑒𝑒 i𝑘𝑘 cos 𝜃𝜃 𝑥𝑥 � 𝜃𝜃 � . The requisite boundary condition at the screen’s surface is thus seen to be satisfied if 𝒥𝒥 is specified as 𝒥𝒥 = − i( 𝐸𝐸 𝜋𝜋𝑍𝑍 ⁄ ) � 𝜃𝜃 . (67) Having found the functional form of 𝒥𝒥 ( 𝜎𝜎 𝑥𝑥 ) , we now turn to the problem of computing the scattered 𝐸𝐸 -field 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) at the arbitrary observation point 𝒓𝒓 = 𝑥𝑥𝒙𝒙� + 𝑧𝑧𝒛𝒛� = 𝑟𝑟 (cos 𝜃𝜃 𝒙𝒙� + sin 𝜃𝜃 𝒛𝒛� ) in accordance with Eqs.(62), (64), and (67); that is, 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) = i( 𝐸𝐸 𝒚𝒚� 𝜋𝜋⁄ ) � �1 + cos 𝜃𝜃 �1+𝜎𝜎 𝑥𝑥 ( 𝜎𝜎 𝑥𝑥 − cos 𝜃𝜃 ) exp[i 𝑘𝑘 ( 𝜎𝜎 𝑥𝑥 𝑥𝑥 + 𝜎𝜎 𝑧𝑧 | 𝑧𝑧 |)] d 𝜎𝜎 𝑥𝑥 ∞−∞ . (68) The integral in Eq.(68) must be evaluated at positive as well as negative values of 𝑥𝑥 , and also for values of 𝑧𝑧 on both sides of the screen. The presence of 𝑘𝑘 𝑧𝑧 in the exponent of the integrand requires that the branch-cuts on both line-segments ( −∞ , − and (1, ∞ ) be taken into account. For these reasons, the integration path of Fig.6(b) ceases to be convenient and we need to change the variable 𝜎𝜎 𝑥𝑥 to something that avoids the need for branch-cuts. We now switch the variable from 𝜎𝜎 𝑥𝑥 to 𝜑𝜑 , where 𝜎𝜎 𝑥𝑥 = cos 𝜑𝜑 , with the integration path in the complex 𝜑𝜑 -plane shown in Fig.7(a). Considering that cos 𝜑𝜑 = cos( 𝜑𝜑 ′ + i 𝜑𝜑 ″ ) = cos 𝜑𝜑 ′ cosh 𝜑𝜑 ″ − i sin 𝜑𝜑 ′ sinh 𝜑𝜑 ″ , (69) the depicted integration path represents the continuous variation of 𝜎𝜎 𝑥𝑥 from −∞ to ∞ . Similarly, 7 𝜎𝜎 𝑧𝑧 = � − cos 𝜑𝜑 = sin 𝜑𝜑 = sin 𝜑𝜑 ′ cosh 𝜑𝜑 ″ + i cos 𝜑𝜑 ′ sinh 𝜑𝜑 ″ (70) is positive real on the horizontal branch, and positive imaginary on both vertical branches of the depicted contour. It is also easy to verify that � − 𝜎𝜎 𝑥𝑥 = √ 𝜑𝜑 ⁄ ) has identical values at corresponding points on the contours of Figs.(6b) and (7a), as does � 𝜎𝜎 𝑥𝑥 = √ 𝜑𝜑 ⁄ ) . Recalling that d 𝜎𝜎 𝑥𝑥 = − sin 𝜑𝜑 d 𝜑𝜑 = −� − cos 𝜑𝜑 d 𝜑𝜑 , we can rewrite Eq.(68) as 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) = − i( 𝐸𝐸 𝒚𝒚� 𝜋𝜋⁄ ) � �1 + cos 𝜃𝜃 �1 − cos 𝜑𝜑 cos 𝜑𝜑 − cos 𝜃𝜃 exp[i 𝑘𝑘 𝑟𝑟 cos( 𝜑𝜑 ∓ 𝜃𝜃 )] d 𝜑𝜑 . (71) The above equation yields the scattered 𝐸𝐸 -field on both sides of the screen, with the minus sign in the exponent corresponding to 𝑧𝑧 > 0 , and the plus sign to 𝑧𝑧 < 0 . In what follows, noting the natural symmetry of the scattered field on the opposite sides of the 𝑥𝑥𝑥𝑥 -plane, we confine our attention to Eq.(71) with only the minus sign in the exponent; the scattered 𝐸𝐸 -field on the left hand side of the screen is subsequently obtained by a simple change of the sign of 𝑧𝑧 . Fig.7 . (a) Contour of integration in the complex 𝜑𝜑 -plane corresponding to the integration path in the 𝜎𝜎 𝑥𝑥 -plane shown in Fig.6(b). By definition, 𝜎𝜎 𝑥𝑥 = cos 𝜑𝜑 , which results in 𝑘𝑘 𝑧𝑧 = 𝑘𝑘 𝜎𝜎 𝑧𝑧 = 𝑘𝑘 sin 𝜑𝜑 , a well-defined function everywhere in the 𝜑𝜑 -plane that does not need branch-cuts. The small bump in the integration path around 𝜑𝜑 = 𝜃𝜃 corresponds to the small semi-circular part of the contour in Fig.6(b). The negative values of 𝜎𝜎 𝑥𝑥″ on the semi-circle translate, in accordance with Eq.(69), into positive value of 𝜑𝜑 ″ on the corresponding bump. (b) Along the steepest-descent contour 𝑆𝑆 ( 𝜃𝜃 ) that passes through the saddle-point 𝜑𝜑 = 𝜃𝜃 , Re[cos( 𝜑𝜑 − 𝜃𝜃 )] is constant. At the saddle-point, 𝑆𝑆 ( 𝜃𝜃 ) makes a angle with the horizontal and vertical lines, which represent contours along which Im[cos( 𝜑𝜑 − 𝜃𝜃 )] is constant. 𝑆𝑆 ( 𝜃𝜃 ) also has the property that Im[cos( 𝜑𝜑 − 𝜃𝜃 )] , which is zero at the saddle-point, rises toward ∞ (continuously and symmetrically on both sides of the saddle) as 𝜑𝜑 moves away from the saddle-point along the contour. The integration path of Fig.7(a) is now replaced with the steepest-descent contour 𝑆𝑆 ( 𝜃𝜃 ) depicted in Fig.7(b). Passing through the saddle-point of exp[i 𝑘𝑘𝑟𝑟 cos( 𝜑𝜑 − 𝜃𝜃 )] at 𝜑𝜑 = 𝜃𝜃 , this contour has the property that Re[cos( 𝜑𝜑 − 𝜃𝜃 )] everywhere on the contour equals 1. In contrast, Im[cos( 𝜑𝜑 − 𝜃𝜃 )] starts at zero at the saddle, then rises toward infinity (continuously and symmetrically on opposite sides of the saddle) as 𝜑𝜑 moves away from the saddle-point at 𝜑𝜑 = 𝜃𝜃 . It is easy to show that the original integration path of Fig.7(a) can be joined to 𝑆𝑆 ( 𝜃𝜃 ) to form a 𝜑𝜑 ′ 𝜋𝜋 𝜑𝜑 ″ 𝜃𝜃 𝜑𝜑 ′ 𝜋𝜋 𝜑𝜑 ″ 𝜃𝜃 𝜃𝜃 𝜃𝜃 + ½ 𝜋𝜋 𝜃𝜃 − ½ 𝜋𝜋 𝑆𝑆 ( 𝜃𝜃 ) (a) (b) ** The only time when the single pole of the integrand at 𝜑𝜑 = 𝜃𝜃 needs to be accounted for is when 𝜃𝜃 < 𝜃𝜃 , at which point the pole is inside the closed contour, as Fig.7(b) clearly indicates. In the vicinity of the pole, the denominator of the integrand in Eq.(71) can be approximated by the first two terms of the Taylor series expansion of cos 𝜑𝜑 , as follows: cos 𝜑𝜑 − cos 𝜃𝜃 = [cos 𝜃𝜃 − sin 𝜃𝜃 ( 𝜑𝜑 − 𝜃𝜃 ) + ⋯ ] − cos 𝜃𝜃 ≅ − sin 𝜃𝜃 ( 𝜑𝜑 − 𝜃𝜃 ) . (72) The residue at the pole is thus seen to be − exp[i 𝑘𝑘 𝑟𝑟 cos( 𝜃𝜃 − 𝜃𝜃 )] , with a corresponding contribution of −𝐸𝐸 𝒚𝒚� exp[i 𝑘𝑘 𝑟𝑟 cos( 𝜃𝜃 − 𝜃𝜃 )] to the scattered 𝐸𝐸 -field. It must be emphasized that the scattered field contributed by the pole at 𝜑𝜑 = 𝜃𝜃 is relevant only when 𝜃𝜃 < 𝜃𝜃 , in which case it cancels the contribution of the incident plane-wave of Eq.(59) to the overall EM field on the right-hand side of the screen, where the screen casts its geometric shadow. Outside this geometric shadow, where 𝜃𝜃 < 𝜃𝜃 ≤ 𝜋𝜋 , the incident beam spills into the 𝑧𝑧 ≥ region — without the counter-balancing effect of the scattered field produced by the pole at 𝜑𝜑 = 𝜃𝜃 . Returning to the scattered 𝐸𝐸 -field produced by the integral in Eq.(71) over the steepest-descent contour 𝑆𝑆 ( 𝜃𝜃 ) , the first term in the integrand can be further streamlined, as follows: �1 +cos 𝜃𝜃 �1 − cos 𝜑𝜑cos 𝜑𝜑 − cos 𝜃𝜃 = ( 𝜃𝜃 ) sin ( 𝜑𝜑 2⁄ ) −2 sin [½( 𝜑𝜑+𝜃𝜃 )] sin [½( 𝜑𝜑−𝜃𝜃 )] = − ½ sin [½( 𝜑𝜑+𝜃𝜃 )] − ½ sin [½( 𝜑𝜑−𝜃𝜃 )] . (73) The scattered 𝐸𝐸 -field on the right-hand side of the screen may thus be written as 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) = i( 𝐸𝐸 𝒚𝒚� 𝜋𝜋⁄ ) 𝑒𝑒 i𝑘𝑘 𝑟𝑟 � � [½( 𝜑𝜑+𝜃𝜃0 )] + [½( 𝜑𝜑−𝜃𝜃0 )] � 𝑆𝑆 ( 𝜃𝜃 ) 𝑒𝑒 i𝑘𝑘 𝑟𝑟 [ cos ( 𝜑𝜑−𝜃𝜃 ) −1 ] d 𝜑𝜑 . (74) Note that we have factored out the imaginary part of the exponent and taken it outside the integral as 𝑒𝑒 i𝑘𝑘 𝑟𝑟 ; what remains of the exponent, namely, i 𝑘𝑘 𝑟𝑟 [cos( 𝜑𝜑 − 𝜃𝜃 ) − , is purely real on 𝑆𝑆 ( 𝜃𝜃 ) . We proceed to compute the integral in Eq.(74) only for the first term of the integrand and refer to it as 𝑬𝑬 𝑠𝑠 , ; the contribution of the second term, 𝑬𝑬 𝑠𝑠 , , will then be found by switching the sign of 𝜃𝜃 . A change of the variable from 𝜑𝜑 to 𝜑𝜑 − 𝜃𝜃 would cause the steepest-descent contour to go through the origin of the 𝜑𝜑 -plane; this shifted contour will now be denoted by 𝑆𝑆 . Taking advantage of the symmetry of 𝑆𝑆 with respect to the origin, we express the final result of integration in terms of the integral on the upper half of 𝑆𝑆 , which is denoted by 𝑆𝑆 + . We will have 𝑬𝑬 𝑠𝑠 , ( 𝒓𝒓 ) = i( 𝐸𝐸 𝒚𝒚� 𝜋𝜋⁄ ) 𝑒𝑒 i𝑘𝑘 𝑟𝑟 � � [½( 𝜑𝜑+𝜃𝜃+𝜃𝜃0 )] + [½( −𝜑𝜑+𝜃𝜃+𝜃𝜃0 )] �𝑒𝑒 i𝑘𝑘0𝑟𝑟 ( cos 𝜑𝜑−1 ) d𝜑𝜑 𝑆𝑆0+ = i( 𝐸𝐸 𝒚𝒚� 𝜋𝜋⁄ ) 𝑒𝑒 i𝑘𝑘 𝑟𝑟 � sin [( 𝜃𝜃+𝜃𝜃0 ) ] cos ( 𝜑𝜑 2⁄ ) cos ( 𝜑𝜑 ) − cos ( 𝜃𝜃+𝜃𝜃0 ) 𝑒𝑒 −i2𝑘𝑘0𝑟𝑟 sin2 ( 𝜑𝜑 2⁄ ) d𝜑𝜑 𝑆𝑆0+ = − i( 𝐸𝐸 𝒚𝒚� 𝜋𝜋⁄ ) sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] 𝑒𝑒 i𝑘𝑘 𝑟𝑟 � cos ( 𝜑𝜑 2⁄ ) exp [ −i2𝑘𝑘0𝑟𝑟 sin2 ( 𝜑𝜑 2⁄ )] sin2 ( 𝜑𝜑 2⁄ ) − sin2 [( 𝜃𝜃+𝜃𝜃0 ) ] d𝜑𝜑 𝑆𝑆0+ . (75) ** On the short line-segments that connect the two contours at 𝜑𝜑 ″ = ± ∞ , the magnitude of the exponential factor in the integrand in Eq.(71) is exp[ 𝑘𝑘 𝑟𝑟 sin( 𝜑𝜑 ′ − 𝜃𝜃 ) sinh( 𝜑𝜑 ″ ) ] . In the upper half-plane, − ≤ sin( 𝜑𝜑 ′ − 𝜃𝜃 ) ≤ and sinh( 𝜑𝜑 ″ ) → ∞ , whereas in the lower half-plane, ≤ sin( 𝜑𝜑 ′ − 𝜃𝜃 ) ≤ and sinh( 𝜑𝜑 ″ ) → −∞ . Thus, in both cases, the integrand vanishes. 𝜑𝜑 to the real-valued 𝜁𝜁 = − exp(i 𝜋𝜋 ⁄ ) sin( 𝜑𝜑 ⁄ ) , where 𝜁𝜁 ranges from to ∞ along the steepest-descent contour 𝑆𝑆 + , now yields †† 𝑬𝑬 𝑠𝑠 , ( 𝒓𝒓 ) = − ( 𝐸𝐸 𝒚𝒚� 𝜋𝜋⁄ ) sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] 𝑒𝑒 i𝑘𝑘 𝑟𝑟 𝑒𝑒 −i𝜋𝜋 4⁄ � exp ( −2𝑘𝑘 𝑟𝑟𝜁𝜁 ) 𝜁𝜁 − i sin [( 𝜃𝜃+𝜃𝜃 ) ] d 𝜁𝜁 ∞0 . (76) The integral appearing in the above equation is evaluated in Appendix G, where it is shown that � exp ( −𝜆𝜆𝜁𝜁 ) 𝜁𝜁 − i𝜂𝜂 d 𝜁𝜁 ∞0 = √𝜋𝜋 | 𝜂𝜂 | −1 𝑒𝑒 −i𝜆𝜆𝜂𝜂 𝐹𝐹 (| 𝜂𝜂 | √𝜆𝜆 ) . (77) Here, 𝐹𝐹 ( 𝛼𝛼 ) = ∫ exp(i 𝑥𝑥 ) d 𝑥𝑥 ∞𝛼𝛼 is the complex Fresnel integral defined in Eq.(46). We thus find 𝑬𝑬 𝑠𝑠 , ( 𝒓𝒓 ) = −�𝐸𝐸 𝒚𝒚� √𝜋𝜋⁄ �𝑒𝑒 −i𝜋𝜋 4⁄ 𝑒𝑒 i𝑘𝑘 𝑟𝑟 𝑒𝑒 −i2𝑘𝑘 𝑟𝑟 sin [( 𝜃𝜃+𝜃𝜃 ) ] 𝐹𝐹�� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] � . (78) The expression for 𝑬𝑬 𝑠𝑠 , ( 𝒓𝒓 ) is derived from Eq.(78) by switching the sign of 𝜃𝜃 . One has to be careful in this case, since sin[( 𝜃𝜃 − 𝜃𝜃 ) 2 ⁄ ] may be negative. Given that both 𝜃𝜃 and 𝜃𝜃 are in the (0, 𝜋𝜋 ) interval, the sign of sin[( 𝜃𝜃 − 𝜃𝜃 ) 2 ⁄ ] will be positive if 𝜃𝜃 > 𝜃𝜃 , and negative if 𝜃𝜃 < 𝜃𝜃 . The scattered 𝐸𝐸 -field of Eq.(74) is thus given by 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) = 𝑬𝑬 𝑠𝑠 , + 𝑬𝑬 𝑠𝑠 , = −�𝐸𝐸 𝒚𝒚� √𝜋𝜋⁄ �𝑒𝑒 −i𝜋𝜋 4⁄ �𝑒𝑒 i𝑘𝑘 𝑟𝑟 cos ( 𝜃𝜃+𝜃𝜃 ) 𝐹𝐹�� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] � ± 𝑒𝑒 i𝑘𝑘 𝑟𝑟 cos ( 𝜃𝜃−𝜃𝜃 ) 𝐹𝐹� ± � 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 − 𝜃𝜃 ) 2 ⁄ ] �� . (79) To this result we must add the contribution of the pole, namely, −𝐸𝐸 𝒚𝒚� exp[i 𝑘𝑘 𝑟𝑟 cos( 𝜃𝜃 − 𝜃𝜃 )] when 𝜃𝜃 < 𝜃𝜃 , and the incident beam 𝐸𝐸 𝒚𝒚� exp[i 𝑘𝑘 𝑟𝑟 cos( 𝜃𝜃 − 𝜃𝜃 )] for the entire range ≤ 𝜃𝜃 ≤ 𝜋𝜋 . Recalling that 𝐹𝐹 ( 𝛼𝛼 ) + 𝐹𝐹 ( −𝛼𝛼 ) = ∫ exp(i 𝑥𝑥 ) d 𝑥𝑥 ∞−∞ = √𝜋𝜋𝑒𝑒 i𝜋𝜋 4⁄ , (80) the total 𝐸𝐸 -field on the right-hand side of the 𝑥𝑥𝑥𝑥 -plane, where ≤ 𝜃𝜃 ≤ 𝜋𝜋 , becomes 𝑬𝑬 total ( 𝒓𝒓 ) = �𝑒𝑒 −i𝜋𝜋 4⁄ 𝐸𝐸 𝒚𝒚� √𝜋𝜋⁄ ��𝑒𝑒 i𝑘𝑘 𝑟𝑟 cos ( 𝜃𝜃−𝜃𝜃 ) 𝐹𝐹�−� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 − 𝜃𝜃 ) 2 ⁄ ] � −𝑒𝑒 i𝑘𝑘 𝑟𝑟 cos ( 𝜃𝜃+𝜃𝜃 ) 𝐹𝐹�� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] �� . (81) On the left-hand side of the screen, where 𝑧𝑧 < 0 and, therefore, −𝜋𝜋 ≤ 𝜃𝜃 ≤ , the scattered 𝐸𝐸 -field is obtained by replacing 𝜃𝜃 with | 𝜃𝜃 | in Eq.(79) as well as in the contribution by the pole at 𝜑𝜑 = 𝜃𝜃 . (Appendix H shows that the scattered 𝐸𝐸 -field in the 𝑧𝑧 < 0 region can also be evaluated by direct integration over a modified contour in the complex 𝜑𝜑 -plane.) Once again, adding the incident 𝐸𝐸 -field and invoking Eq.(80), we find 𝑬𝑬 total ( 𝒓𝒓 ) = �𝑒𝑒 −i𝜋𝜋 4⁄ 𝐸𝐸 𝒚𝒚� √𝜋𝜋⁄ ��𝑒𝑒 i𝑘𝑘 𝑟𝑟 cos ( 𝜃𝜃−𝜃𝜃 ) 𝐹𝐹�� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 − 𝜃𝜃 ) 2 ⁄ ] � −𝑒𝑒 i𝑘𝑘 𝑟𝑟 cos ( 𝜃𝜃+𝜃𝜃 ) 𝐹𝐹�−� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] �� . (82) One obtains Eq.(82) by replacing 𝜃𝜃 in Eq.(81) with 𝜋𝜋 + 𝜃𝜃 , the latter 𝜃𝜃 being in the ( −𝜋𝜋 , 0) interval. Thus, Eq.(81) with ≤ 𝜃𝜃 ≤ 𝜋𝜋 covers the entire range of observation points 𝒓𝒓 . A clear †† Upon setting 𝜁𝜁 = i sin ( 𝜑𝜑 ⁄ ) , we find two possible choices for the new variable, namely, 𝜁𝜁 = ± 𝑒𝑒 i𝜋𝜋 4⁄ sin( 𝜑𝜑 ⁄ ) . Of these, the one with the minus sign represents the upper-half 𝑆𝑆 + of the steepest-descent trajectory. To see this, note that as 𝑆𝑆 + approaches its extremity when 𝜑𝜑 → − ½ 𝜋𝜋 + i ∞ , we have sin( 𝜑𝜑 ⁄ ) → ( − 𝜑𝜑 ″ ⁄ ) 2 √ ⁄ , which must be multiplied by −𝑒𝑒 i𝜋𝜋 4⁄ for 𝜁𝜁 to approach + ∞ . 𝐹𝐹 ( 𝛼𝛼 ) ; Appendix F contains a detailed explanation in terms of the Cornu spiral representation of 𝐹𝐹 ( 𝛼𝛼 ) . ‡‡ The general expression for the total (i.e., incident plus scattered) 𝐸𝐸 -field given in Eq.(81) is somewhat simplified in terms of the new function Φ ( 𝛼𝛼 ) = 𝑒𝑒 −i𝛼𝛼 𝐹𝐹 ( 𝛼𝛼 ) , as follows: 𝑬𝑬 total ( 𝒓𝒓 ) = 𝐸𝐸 𝒚𝒚�𝑒𝑒 i𝑘𝑘0𝑟𝑟 √𝜋𝜋𝑒𝑒 i𝜋𝜋 4⁄ Φ�−� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 − 𝜃𝜃 ) 2 ⁄ ] � − Φ�� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] � ; (0 ≤ 𝜃𝜃 ≤ 𝜋𝜋 ) . (83) It is worth mentioning that, the contribution to the scattered 𝐸𝐸 -field by the pole at 𝜑𝜑 = 𝜃𝜃 , namely, 𝑬𝑬 𝑠𝑠 , ( 𝒓𝒓 ) = −𝐸𝐸 𝒚𝒚� exp[i 𝑘𝑘 𝑟𝑟 cos( 𝜃𝜃 − 𝜃𝜃 )] , which exists only in the intervals ≤ 𝜃𝜃 < 𝜃𝜃 and 𝜋𝜋 − 𝜃𝜃 < 𝜃𝜃 ≤ 𝜋𝜋 , cancels the incident 𝐸𝐸 -field in the shadow region behind the screen, while acting as the reflected field in front of the perfectly conducting half-mirror. . The total 𝐻𝐻 -field is computed from the 𝐸𝐸 -field of Eq.(83) with the aid of Maxwell’s equation i 𝑘𝑘 𝑍𝑍 𝑯𝑯 ( 𝒓𝒓 ) = 𝜵𝜵 × 𝑬𝑬 ( 𝒓𝒓 ) = − ( 𝑟𝑟 −1 𝜕𝜕 𝜃𝜃 𝐸𝐸 𝑦𝑦 ) 𝒓𝒓� + ( 𝜕𝜕 𝑟𝑟 𝐸𝐸 𝑦𝑦 ) 𝜽𝜽� . The identity Φ ′ ( 𝛼𝛼 ) = − − 𝛼𝛼Φ ( 𝛼𝛼 ) will be used in this calculation. To simplify the notation, we introduce the new variables 𝜉𝜉 = −� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 − 𝜃𝜃 ) 2 ⁄ ] and 𝜉𝜉 = � 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] . We find 𝐻𝐻 𝑟𝑟 ( 𝒓𝒓 ) = 𝐸𝐸 𝑒𝑒 i𝑘𝑘0𝑟𝑟 √𝜋𝜋 𝑒𝑒 i𝜋𝜋 4⁄ 𝑍𝑍 �Φ ( 𝜉𝜉 ) sin( 𝜃𝜃 − 𝜃𝜃 ) − Φ ( 𝜉𝜉 ) sin( 𝜃𝜃 + 𝜃𝜃 ) + i√2 cos ( 𝜃𝜃 2⁄ ) cos ( 𝜃𝜃 ) �𝑘𝑘 𝑟𝑟 � . (84) 𝐻𝐻 𝜃𝜃 ( 𝒓𝒓 ) = 𝐸𝐸 𝑒𝑒 i𝑘𝑘0𝑟𝑟 √𝜋𝜋 𝑒𝑒 i𝜋𝜋 4⁄ 𝑍𝑍 �Φ ( 𝜉𝜉 ) cos( 𝜃𝜃 − 𝜃𝜃 ) − Φ ( 𝜉𝜉 ) cos( 𝜃𝜃 + 𝜃𝜃 ) − i√2 sin ( 𝜃𝜃 2⁄ ) cos ( 𝜃𝜃 ) �𝑘𝑘 𝑟𝑟 � . (85) The Cartesian components 𝐻𝐻 𝑥𝑥 = 𝐻𝐻 𝑟𝑟 cos 𝜃𝜃 − 𝐻𝐻 𝜃𝜃 sin 𝜃𝜃 and 𝐻𝐻 𝑧𝑧 = 𝐻𝐻 𝑟𝑟 sin 𝜃𝜃 + 𝐻𝐻 𝜃𝜃 cos 𝜃𝜃 of the magnetic field may now be obtained from the polar components 𝐻𝐻 𝑟𝑟 and 𝐻𝐻 𝜃𝜃 , as follows: 𝐻𝐻 𝑥𝑥 ( 𝒓𝒓 ) = − 𝐸𝐸 𝑒𝑒 i𝑘𝑘0𝑟𝑟 √𝜋𝜋 𝑒𝑒 i𝜋𝜋 4⁄ 𝑍𝑍 � [ Φ ( 𝜉𝜉 ) + Φ ( 𝜉𝜉 )] sin 𝜃𝜃 − i√2 cos ( 𝜃𝜃 2⁄ ) cos ( 𝜃𝜃 ) �𝑘𝑘 𝑟𝑟 � . (86) 𝐻𝐻 𝑧𝑧 ( 𝒓𝒓 ) = + 𝐸𝐸 𝑒𝑒 i𝑘𝑘0𝑟𝑟 √𝜋𝜋 𝑒𝑒 i𝜋𝜋 4⁄ 𝑍𝑍 � [ Φ ( 𝜉𝜉 ) − Φ ( 𝜉𝜉 )] cos 𝜃𝜃 + i√2 sin ( 𝜃𝜃 2⁄ ) cos ( 𝜃𝜃 ) �𝑘𝑘 𝑟𝑟 � . (87) It is readily verified that 𝐻𝐻 𝑧𝑧 = 0 at the surface of the conductor, where 𝜃𝜃 = 0 , and that 𝐻𝐻 𝑥𝑥 equals the 𝑥𝑥 component of the incident 𝐻𝐻 -field in the open half of the 𝑥𝑥𝑥𝑥 -plane, where 𝜃𝜃 = 𝜋𝜋 .
8. Far field scattering and the optical theorem . In the system of Fig.1(b), let the object inside the closed surface 𝑆𝑆 be illuminated from the outside, and let the observation point 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 be far away from the object, so that the approximate form of 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) given in Eq.(50) along with its corresponding gradient 𝜵𝜵𝐺𝐺 ≅ − i 𝑘𝑘 𝝈𝝈 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) would be applicable. Denoting by 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) and 𝑩𝑩 𝑠𝑠 ( 𝒓𝒓 ) the scattered fields appearing on 𝑆𝑆 , Eq.(28) yields the 𝐸𝐸 -field at the observation point as 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) ≅ i𝑘𝑘 exp ( i𝑘𝑘 𝑟𝑟 ) ∫ [ 𝑐𝑐 ( 𝒏𝒏� × 𝑩𝑩 𝑠𝑠 ) − ( 𝒏𝒏� × 𝑬𝑬 𝑠𝑠 ) × 𝝈𝝈 − ( 𝒏𝒏� ∙ 𝑬𝑬 𝑠𝑠 ) 𝝈𝝈 ]e −i𝑘𝑘 𝝈𝝈 ∙ 𝒓𝒓 d 𝑠𝑠 𝑆𝑆1 . (88) Considering that the local field in the vicinity of 𝒓𝒓 has the character of a plane-wave, the last term in the above integrand, which represents a contribution to 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) that is aligned with the local 𝑘𝑘 -vector, is expected to be cancelled out by an equal but opposite contribution from the first term. We thus arrive at the following simplified version of Eq.(88): ‡‡ Our Eq.(81) agrees with the corresponding result in Born & Wolf’s
Principles of Optics , provided that the angles 𝜃𝜃 and 𝛼𝛼 in their Eq.(22) of Chapter 11, Section 5, are recognized as 𝜋𝜋 − 𝜃𝜃 and 𝜋𝜋 − 𝜃𝜃 in our notation. 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) ≅ − i𝑘𝑘 exp ( i𝑘𝑘 𝑟𝑟 ) 𝝈𝝈 × ∫ [ 𝑐𝑐𝝈𝝈 × ( 𝒏𝒏� × 𝑩𝑩 𝑠𝑠 ) − 𝒏𝒏� × 𝑬𝑬 𝑠𝑠 ]e −i𝑘𝑘 𝝈𝝈 ∙ 𝒓𝒓 d 𝑠𝑠 𝑆𝑆1 . (89) With reference to Fig.8, suppose now that the object is illuminated (and thus excited) by a plane-wave arriving along the unit-vector 𝝈𝝈 inc , whose 𝑬𝑬 and 𝑩𝑩 fields are 𝑬𝑬 inc ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑬𝑬 i exp[i 𝑘𝑘 ( 𝝈𝝈 inc ∙ 𝒓𝒓 − 𝑐𝑐𝑡𝑡 )] . (90) 𝑩𝑩 inc ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑐𝑐 −1 𝝈𝝈 inc × 𝑬𝑬 i exp[i 𝑘𝑘 ( 𝝈𝝈 inc ∙ 𝒓𝒓 − 𝑐𝑐𝑡𝑡 )] . (91) The time-averaged total Poynting vector on the surface 𝑆𝑆 is readily evaluated, as follows: 〈𝑺𝑺 total ( 𝒓𝒓 ) 〉 = ½Re � ( 𝑬𝑬 i 𝑒𝑒 i𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 + 𝑬𝑬 𝑠𝑠 ) × 𝜇𝜇 −1 ( 𝑩𝑩 i 𝑒𝑒 i𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 + 𝑩𝑩 𝑠𝑠 ) ∗� = ½ 𝜇𝜇 −1 Re �𝑬𝑬 i × 𝑩𝑩 i ∗ + 𝑬𝑬 s × 𝑩𝑩 s ∗ + 𝑬𝑬 i 𝑒𝑒 i𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 × 𝑩𝑩 s ∗ + 𝑬𝑬 s × 𝑩𝑩 i ∗ 𝑒𝑒 −i𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 � = ½ 𝑍𝑍 −1 ( 𝑬𝑬 i ∙ 𝑬𝑬 i ∗ ) 𝝈𝝈 inc + ½ 𝜇𝜇 −1 Re( 𝑬𝑬 s × 𝑩𝑩 s ∗ ) +½ 𝑍𝑍 −1 Re � [ 𝑐𝑐𝑬𝑬 i ∗ × 𝑩𝑩 s + 𝑬𝑬 s × ( 𝝈𝝈 inc × 𝑬𝑬 i ∗ )] 𝑒𝑒 −i𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 � . (92) If we dot-multiply both sides of Eq.(92) by −𝒏𝒏� , then integrate over the closed surface 𝑆𝑆 , we obtain, on the left-hand side, the total rate of the inward flow of EM energy, which is the absorbed EM power by the object. On the right-hand side, the first term integrates to zero, because ( 𝑬𝑬 i ∙ 𝑬𝑬 i ∗ ) 𝝈𝝈 inc is a constant and ∮ 𝒏𝒏� d 𝑠𝑠 𝑆𝑆1 = 0 . The integral of the second term will be the negative time-rate of the energy departure from the object via scattering, which can be moved to the left-hand side of the equation. The combination of the two terms on the left-hand side now yields the total EM power that is taken away from the incident beam — either by absorption or due to scattering. We will have
Absorbed + Scattered Power = − ½ 𝑍𝑍 −1 Re ∮ 𝒏𝒏� ∙ [ 𝑐𝑐𝑬𝑬 i ∗ × 𝑩𝑩 s + 𝑬𝑬 s × ( 𝝈𝝈 inc × 𝑬𝑬 i ∗ )] 𝑒𝑒 −i𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 d 𝑠𝑠 𝑆𝑆1 = ½ 𝑍𝑍 −1 Re ∮ [ 𝑐𝑐 ( 𝒏𝒏� × 𝑩𝑩 s ) ∙ 𝑬𝑬 i ∗ − ( 𝒏𝒏� × 𝑬𝑬 s ) ∙ ( 𝝈𝝈 inc × 𝑬𝑬 i ∗ )] 𝑒𝑒 −i𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 d 𝑠𝑠 𝑆𝑆1 = ½ 𝑍𝑍 −1 Re �𝑬𝑬 i ∗ ∙ ∮ [ 𝑐𝑐 ( 𝒏𝒏� × 𝑩𝑩 s ) − ( 𝒏𝒏� × 𝑬𝑬 s ) × 𝝈𝝈 inc ] 𝑒𝑒 −i𝑘𝑘 𝝈𝝈 inc ∙ 𝒓𝒓 d 𝑠𝑠 𝑆𝑆1 � . (93) Fig.8 . A monochromatic plane-wave propagating along the unit-vector 𝝈𝝈 inc is scattered from a small object in the vicinity of the origin of the coordinate system. The scattered electric and magnetic fields on the closed surface 𝑆𝑆 surrounding the object are denoted by 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) and 𝑩𝑩 𝑠𝑠 ( 𝒓𝒓 ) . The surface normals 𝒏𝒏� at every point on 𝑆𝑆 are outward directed. The scattered light reaching the far away observation point 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 has the 𝑘𝑘 -vector 𝑘𝑘 𝝈𝝈 and the EM fields 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 ) and 𝑩𝑩 𝑠𝑠 ( 𝒓𝒓 ) . 𝑧𝑧 𝑥𝑥 𝒏𝒏 � 𝒓𝒓 𝝈𝝈 × 𝑆𝑆 𝒏𝒏 � 𝝈𝝈 inc 𝑍𝑍 = �𝜇𝜇 𝜀𝜀 ⁄ ≅ Ω 𝒂𝒂 ∙ ( 𝒃𝒃 × 𝒄𝒄 ) = ( 𝒂𝒂 × 𝒃𝒃 ) ∙ 𝒄𝒄 𝐸𝐸 -field in the direction of 𝝈𝝈 = 𝝈𝝈 inc as observed in the far field. (Recall that the term ( 𝒏𝒏� ∙ 𝑬𝑬 𝑠𝑠 ) 𝝈𝝈 in the integrand of Eq.(88) has been deemed inconsequential.) If Eq.(93) is normalized by the incident EM power per unit area, namely, 𝑃𝑃 inc = ½ 𝑍𝑍 −1 Re( 𝑬𝑬 i ∙ 𝑬𝑬 i ∗ ) , the left-hand side will become the scattering cross-section of the object, while the right-hand side, aside from the coefficient i 𝑘𝑘 𝑒𝑒 i𝑘𝑘 𝑟𝑟 (4 𝜋𝜋𝑟𝑟 ) ⁄ , will be the projection of the forward-scattered 𝐸𝐸 -field (i.e., 𝝈𝝈 = 𝝈𝝈 inc ) on the incident 𝐸𝐸 -field. This important result in the theory of scattering has come to be known as the optical theorem (or the optical cross-section theorem).
9. Scattering from weak inhomogeneities . Figure 9 shows a monochromatic plane-wave of frequency 𝜔𝜔 passing through a transparent, linear, isotropic medium that has a region of weak inhomogeneities in the vicinity of the origin of coordinates. The host medium is described by its relative permittivity 𝜀𝜀 ( 𝒓𝒓 , 𝜔𝜔 ) and permeability 𝜇𝜇 ( 𝒓𝒓 , 𝜔𝜔 ) , which consist of a spatially homogeneous background plus slight variations (localized in the vicinity of 𝒓𝒓 = 0 ) on this background; that is, 𝜀𝜀 ( 𝒓𝒓 , 𝜔𝜔 ) = 𝜀𝜀 ℎ ( 𝜔𝜔 ) + 𝛿𝛿𝜀𝜀 ( 𝒓𝒓 , 𝜔𝜔 ) . (94) 𝜇𝜇 ( 𝒓𝒓 , 𝜔𝜔 ) = 𝜇𝜇 ℎ ( 𝜔𝜔 ) + 𝛿𝛿𝜇𝜇 ( 𝒓𝒓 , 𝜔𝜔 ) . (95) The displacement field is thus written as 𝑫𝑫 ( 𝒓𝒓 , 𝜔𝜔 ) = 𝜀𝜀 𝜀𝜀 ( 𝒓𝒓 , 𝜔𝜔 ) 𝑬𝑬 ( 𝒓𝒓 , 𝜔𝜔 ) and the magnetic induction is given by 𝑩𝑩 ( 𝒓𝒓 , 𝜔𝜔 ) = 𝜇𝜇 𝜇𝜇 ( 𝒓𝒓 , 𝜔𝜔 ) 𝑯𝑯 ( 𝒓𝒓 , 𝜔𝜔 ) . In the absence of free charges and currents, i.e., when 𝜌𝜌 free ( 𝒓𝒓 , 𝜔𝜔 ) = 0 and 𝑱𝑱 free ( 𝒓𝒓 , 𝜔𝜔 ) = 0 , Maxwell’s macroscopic equations will be 𝜵𝜵 ∙ 𝑫𝑫 = 0 ; 𝜵𝜵 × 𝑯𝑯 = − i 𝜔𝜔𝑫𝑫 ; 𝜵𝜵 × 𝑬𝑬 = i 𝜔𝜔𝑩𝑩 ; 𝜵𝜵 ∙ 𝑩𝑩 = 0 . (96) Noting that the 𝑫𝑫 and 𝑩𝑩 fields depart only slightly from the respective values 𝜀𝜀 𝜀𝜀 ℎ 𝑬𝑬 and 𝜇𝜇 𝜇𝜇 ℎ 𝑯𝑯 that they would have had in the absence of the 𝛿𝛿𝜀𝜀 and 𝛿𝛿𝜇𝜇 perturbations, we write 𝜵𝜵 × 𝜵𝜵 × ( 𝑫𝑫 − 𝜀𝜀 𝜀𝜀 ℎ 𝑬𝑬 ) = 𝜵𝜵 ( 𝜵𝜵 ∙ 𝑫𝑫 ) − 𝜵𝜵 𝑫𝑫 − i 𝜔𝜔𝜀𝜀 𝜀𝜀 ℎ 𝜵𝜵 × 𝑩𝑩 = −𝜵𝜵 𝑫𝑫 − i 𝜔𝜔𝜀𝜀 𝜀𝜀 ℎ [ 𝜵𝜵 × ( 𝑩𝑩 − 𝜇𝜇 𝜇𝜇 ℎ 𝑯𝑯 ) + 𝜇𝜇 𝜇𝜇 ℎ 𝜵𝜵 × 𝑯𝑯 ] = −𝜵𝜵 𝑫𝑫 − i 𝜔𝜔𝜀𝜀 𝜀𝜀 ℎ 𝜵𝜵 × ( 𝑩𝑩 − 𝜇𝜇 𝜇𝜇 ℎ 𝑯𝑯 ) − ( 𝜔𝜔 𝑐𝑐⁄ ) 𝜇𝜇 ℎ 𝜀𝜀 ℎ 𝑫𝑫 . (97) Fig.9 . A monochromatic plane-wave of frequency 𝜔𝜔 and 𝐸𝐸 -field amplitude 𝑬𝑬 inc propagates along the unit-vector 𝝈𝝈 inc within a mildly inhomogeneous host medium of refractive index 𝑛𝑛 ℎ ( 𝜔𝜔 ) =( 𝜇𝜇 ℎ 𝜀𝜀 ℎ ) ½ . The inhomogeneous region of the host, a small patch in the vicinity of the origin of coordinates, is specified by its relative permittivity 𝜀𝜀 ℎ ( 𝜔𝜔 ) + 𝛿𝛿𝜀𝜀 ( 𝒓𝒓 , 𝜔𝜔 ) and relative permeability 𝜇𝜇 ℎ ( 𝜔𝜔 ) + 𝛿𝛿𝜇𝜇 ( 𝒓𝒓 , 𝜔𝜔 ) . The scattered field is observed at the faraway point 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 . 𝑧𝑧 𝑥𝑥 𝒓𝒓 𝝈𝝈 × 𝝈𝝈 inc ( 𝜀𝜀 ℎ , 𝜇𝜇 ℎ ) ( 𝛿𝛿𝜀𝜀 , 𝛿𝛿𝜇𝜇 ) 𝑛𝑛 ℎ = ( 𝜇𝜇 ℎ 𝜀𝜀 ℎ ) ½ , and that the free-space wave-number is 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ , Eq.(97) may be rewritten as 𝜵𝜵 𝑫𝑫 + ( 𝑘𝑘 𝑛𝑛 ℎ ) 𝑫𝑫 = − i( 𝑘𝑘 𝜀𝜀 ℎ 𝑐𝑐⁄ ) 𝜵𝜵 × ( 𝛿𝛿𝜇𝜇𝑯𝑯 ) − 𝜀𝜀 𝜵𝜵 × 𝜵𝜵 × ( 𝛿𝛿𝜀𝜀𝑬𝑬 ) . (98) This Helmholtz equation has a homogeneous solution, which we denote by 𝑫𝑫 ℎ ( 𝒓𝒓 , 𝜔𝜔 ) , and an inhomogeneous solution, which arises from the local deviations 𝛿𝛿𝜀𝜀 and 𝛿𝛿𝜇𝜇 of the host medium. Recalling that the Green function 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) = exp(i 𝑘𝑘 𝑛𝑛 ℎ | 𝒓𝒓 − 𝒓𝒓 |) | 𝒓𝒓 − 𝒓𝒓 | ⁄ is a solution of the Helmholtz equation 𝜵𝜵 𝐺𝐺 + ( 𝑘𝑘 𝑛𝑛 ℎ ) 𝐺𝐺 = − 𝜋𝜋𝛿𝛿 ( 𝒓𝒓 − 𝒓𝒓 ) , an integral relation for the scattered field solution 𝑫𝑫 𝑠𝑠 ( 𝒓𝒓 , 𝜔𝜔 ) of Eq.(98) in terms of the total fields 𝑬𝑬 ( 𝒓𝒓 , 𝜔𝜔 ) and 𝑯𝑯 ( 𝒓𝒓 , 𝜔𝜔 ) will be 𝑫𝑫 𝑠𝑠 ( 𝒓𝒓 , 𝜔𝜔 ) = (4 𝜋𝜋 ) −1 ∫ [i( 𝑘𝑘 𝜀𝜀 ℎ 𝑐𝑐⁄ ) 𝜵𝜵 × ( 𝛿𝛿𝜇𝜇𝑯𝑯 ) + 𝜀𝜀 𝜵𝜵 × 𝜵𝜵 × ( 𝛿𝛿𝜀𝜀𝑬𝑬 )] volume 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝒓𝒓 . (99) Using a far-field approximation to 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) similar to that in Eq.(50), we will have 𝑫𝑫 𝑠𝑠 ( 𝒓𝒓 , 𝜔𝜔 ) ≅ exp ( i𝑘𝑘 𝑛𝑛 ℎ 𝑟𝑟 ) ∫ [i( 𝑘𝑘 𝜀𝜀 ℎ 𝑐𝑐⁄ ) 𝜵𝜵 × ( 𝛿𝛿𝜇𝜇𝑯𝑯 ) + 𝜀𝜀 𝜵𝜵 × 𝜵𝜵 × ( 𝛿𝛿𝜀𝜀𝑬𝑬 )] 𝑒𝑒 −i𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 ∙ 𝒓𝒓 d 𝒓𝒓 volume . (100) The vector identity ( 𝜵𝜵 × 𝑽𝑽 ) 𝑒𝑒 −i𝒌𝒌 ∙ 𝒓𝒓 = i 𝒌𝒌 × 𝑽𝑽𝑒𝑒 −i𝒌𝒌 ∙ 𝒓𝒓 + 𝜵𝜵 × ( 𝑽𝑽𝑒𝑒 −i𝒌𝒌 ∙ 𝒓𝒓 ) can be used to replace the first term in the integrand of Eq.(100) with − ( 𝑘𝑘 𝜀𝜀 ℎ 𝑛𝑛 ℎ 𝑐𝑐⁄ ) 𝑒𝑒 −i𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 ∙ 𝒓𝒓 𝝈𝝈 × 𝛿𝛿𝜇𝜇𝑯𝑯 . The volume integral of 𝜵𝜵 × ( 𝑽𝑽𝑒𝑒 −i𝒌𝒌 ∙ 𝒓𝒓 ) becomes the surface integral of 𝑒𝑒 −i𝒌𝒌 ∙ 𝒓𝒓 𝑽𝑽 × d 𝒔𝒔 , which subsequently vanishes because, for away from the inhomogeneous region, 𝛿𝛿𝜇𝜇 → . Similarly, the second term of the integrand is replaced by i 𝜀𝜀 𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 × [ 𝜵𝜵 × ( 𝛿𝛿𝜀𝜀𝑬𝑬 )] 𝑒𝑒 −i𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 ∙ 𝒓𝒓 . Another application of the vector identity then replaces the remaining term with −𝜀𝜀 𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 × ( 𝝈𝝈 × 𝛿𝛿𝜀𝜀𝑬𝑬 ) 𝑒𝑒 −i𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 ∙ 𝒓𝒓 . We thus arrive at 𝑫𝑫 𝑠𝑠 ( 𝒓𝒓 , 𝜔𝜔 ) ≅ ( 𝑘𝑘 𝑛𝑛 ℎ ) exp ( i𝑘𝑘 𝑛𝑛 ℎ 𝑟𝑟 ) ∫ [( 𝜀𝜀 ℎ 𝑛𝑛 ℎ 𝑐𝑐⁄ ) 𝛿𝛿𝜇𝜇𝑯𝑯 + 𝜀𝜀 𝝈𝝈 × 𝛿𝛿𝜀𝜀𝑬𝑬 ] × 𝝈𝝈 𝑒𝑒 −i𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 ∙ 𝒓𝒓 d 𝒓𝒓 volume . (101) In the first Born approximation, the 𝑬𝑬 ( 𝒓𝒓 ) and 𝑯𝑯 ( 𝒓𝒓 ) fields in the integrand of Eq.(101) are replaced with the solutions 𝑬𝑬 ℎ ( 𝒓𝒓 ) and 𝑯𝑯 ℎ ( 𝒓𝒓 ) of the homogeneous Helmholtz equation. When the homogeneous background wave is a plane-wave, we will have 𝑬𝑬 ℎ ( 𝒓𝒓 ) = 𝑬𝑬 inc exp(i 𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 inc ∙ 𝒓𝒓 ) . (102) 𝑯𝑯 ℎ ( 𝒓𝒓 ) = � 𝜀𝜀 𝜀𝜀 ℎ 𝜇𝜇 𝜇𝜇 ℎ 𝝈𝝈 inc × 𝑬𝑬 inc exp(i 𝑘𝑘 𝑛𝑛 ℎ 𝝈𝝈 inc ∙ 𝒓𝒓 ) . (103) A final substitution from Eqs.(102) and (103) into Eq.(101) yields 𝑫𝑫 𝑠𝑠 ( 𝒓𝒓 , 𝜔𝜔 ) ≅ 𝜀𝜀 𝜀𝜀 ℎ 𝑘𝑘 exp ( i𝑘𝑘 𝑛𝑛 ℎ 𝑟𝑟 ) ∫ [( 𝜀𝜀 ℎ 𝛿𝛿𝜇𝜇𝝈𝝈 inc + 𝜇𝜇 ℎ 𝛿𝛿𝜀𝜀𝝈𝝈 ) × 𝑬𝑬 inc ] volume × 𝝈𝝈 𝑒𝑒 i𝑘𝑘 𝑛𝑛 ℎ ( 𝝈𝝈 inc − 𝝈𝝈 ) ∙ 𝒓𝒓 d 𝒓𝒓 . (104) Thus, in the first Born approximation, the scattered field 𝑫𝑫 𝑠𝑠 ( 𝒓𝒓 , 𝜔𝜔 ) = 𝜀𝜀 𝜀𝜀 ℎ 𝑬𝑬 𝑠𝑠 ( 𝒓𝒓 , 𝜔𝜔 ) is directly related to the host medium perturbations 𝛿𝛿𝜀𝜀 ( 𝒓𝒓 , 𝜔𝜔 ) and 𝛿𝛿𝜇𝜇 ( 𝒓𝒓 , 𝜔𝜔 ) via the volume integral in Eq.(104). Here, 𝑬𝑬 inc embodies not only the strength but also the polarization state of the incident plane-wave, the unit-vector 𝝈𝝈 inc is the direction of incidence, 𝝈𝝈 = 𝒓𝒓 / 𝑟𝑟 is a unit-vector pointing from the origin of coordinates to the observation point 𝒓𝒓 , and 𝒒𝒒 = 𝑘𝑘 𝑛𝑛 ℎ ( 𝝈𝝈 inc − 𝝈𝝈 ) is the difference between the incident and scattered 𝑘𝑘 -vectors. d 𝒓𝒓 stands for d 𝑥𝑥 d 𝑥𝑥 d 𝑧𝑧
10. Neutron scattering from magnetic electrons in Born’s first approximation . The scattering of slow neutrons from ferromagnetic materials can be treated in ways that are similar to our analysis of EM scattering from mild inhomogeneities discussed in the preceding section. The wave function 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) of a particle of mass 𝓂𝓂 in the scalar potential field 𝑉𝑉 ( 𝒓𝒓 , 𝑡𝑡 ) satisfies the following Schrödinger equation: i ℏ𝜕𝜕 𝑡𝑡 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) = − ( ℏ 𝓂𝓂⁄ ) 𝛻𝛻 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) + 𝑉𝑉 ( 𝒓𝒓 , 𝑡𝑡 ) 𝜓𝜓 ( 𝒓𝒓 , 𝑡𝑡 ) . (105) When the potential is time-independent and the particle is in an eigenstate of energy ℰ 𝑛𝑛 , we will have the time-independent Schrödinger equation, namely, [( ℏ 𝓂𝓂⁄ ) 𝛻𝛻 + ℰ 𝑛𝑛 ] 𝜓𝜓 ( 𝒓𝒓 ) = 𝑉𝑉 ( 𝒓𝒓 ) 𝜓𝜓 ( 𝒓𝒓 ) . (106) The Green function for Eq.(106) is 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ′ ) = 𝑒𝑒 i𝑘𝑘 | 𝒓𝒓 – 𝒓𝒓 ′ | | 𝒓𝒓 − 𝒓𝒓 ′ | � , where ℏ𝑘𝑘 = (2 𝓂𝓂ℰ 𝑛𝑛 ) ½ . If, in the absence of the potential 𝑉𝑉 ( 𝒓𝒓 ) , the solution of the homogeneous equation is found to be 𝜓𝜓 ( 𝒓𝒓 ) , then, when the potential is introduced, we will have 𝜓𝜓 ( 𝒓𝒓 ) = 𝜓𝜓 ( 𝒓𝒓 ) − ( 𝓂𝓂 𝜋𝜋ℏ ⁄ ) ∭ 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ′ ) 𝑉𝑉 ( 𝒓𝒓 ′ ) 𝜓𝜓 ( 𝒓𝒓 ′ )d 𝒓𝒓 ′∞−∞ . (107) Note that Eq.(107) is not an actual solution of Eq.(106); rather, considering that the desired wave-function 𝜓𝜓 ( 𝒓𝒓 ) appears in the integrand on the right hand-side, Eq.(107) is an integral form of the differential equation (106). In the case of Born’s first approximation, one assumes that 𝑉𝑉 ( 𝒓𝒓 ) is a fairly weak potential, in which case 𝜓𝜓 ( 𝒓𝒓 ′ ) can be substituted for 𝜓𝜓 ( 𝒓𝒓 ′ ) , yielding 𝜓𝜓 ( 𝒓𝒓 ) ≅ 𝜓𝜓 ( 𝒓𝒓 ) − ( 𝓂𝓂 𝜋𝜋ℏ ⁄ ) ∭ 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ′ ) 𝑉𝑉 ( 𝒓𝒓 ′ ) 𝜓𝜓 ( 𝒓𝒓 ′ )d 𝒓𝒓 ′∞−∞ . (108) In a typical scattering problem, an incoming particle of mass 𝓂𝓂 and well-defined momentum 𝒑𝒑 = ℏ𝒌𝒌 has the initial wave-function 𝜓𝜓 ( 𝒓𝒓 ) = 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 . Upon interacting with a weak scattering potential 𝑉𝑉 ( 𝒓𝒓 ) , the wave-function will change in accordance with Eq.(108). Let 𝑉𝑉 ( 𝒓𝒓 ) have significant values only in the vicinity of the origin, 𝒓𝒓 = 0 , and assume that the scattering process is elastic, so that the momentum 𝒑𝒑� of the particle after scattering will have the same magnitude ℏ𝑘𝑘 as before, but the direction of propagation changes from that of 𝒌𝒌 to that of 𝒌𝒌� . At an observation point 𝒓𝒓 far from the origin, that is, | 𝒓𝒓 | ≫ | 𝒓𝒓 ′ | , the scattered particle’s momentum is expected to be 𝒑𝒑� = ℏ𝒌𝒌� = ℏ𝑘𝑘𝒓𝒓� , and the Green function may be approximated as exp ( i𝑘𝑘 | 𝒓𝒓 – 𝒓𝒓 ′ |)| 𝒓𝒓 – 𝒓𝒓 ′ | ≅ exp [ i𝑘𝑘� ( 𝒓𝒓 − 𝒓𝒓 ′ ) ∙ ( 𝒓𝒓 − 𝒓𝒓 ′ )] 𝑟𝑟 ≅ exp [ i𝑘𝑘 ( 𝑟𝑟 − 𝒓𝒓 ′ ∙ 𝒓𝒓� )] 𝑟𝑟 = exp ( i𝑘𝑘𝑟𝑟 ) 𝑟𝑟 𝑒𝑒 −i𝒌𝒌� ∙ 𝒓𝒓 ′ . (109) Substitution into Eq.(108) now yields 𝜓𝜓 ( 𝒓𝒓 ) ≅ 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 − � 𝓂𝓂2𝜋𝜋ℏ � 𝑒𝑒 i𝑘𝑘𝑟𝑟 𝑟𝑟 � 𝑉𝑉 ( 𝒓𝒓 ′ ) 𝑒𝑒 i ( 𝒌𝒌 – 𝒌𝒌� ) ∙ 𝒓𝒓 ′ d 𝒓𝒓 ′ ∞−∞ . (110) Denoting the change in the direction of the particle’s momentum by 𝒒𝒒 = 𝒑𝒑 − 𝒑𝒑� , and noting that 𝑒𝑒 i𝑘𝑘𝑟𝑟 𝑟𝑟⁄ is simply a spherical wave emanating from the origin, the scattering amplitude 𝑓𝑓 ( 𝒒𝒒 ) is readily seen to be 𝑓𝑓 ( 𝒒𝒒 ) ≅ − 𝓂𝓂2𝜋𝜋ℏ � 𝑉𝑉 ( 𝒓𝒓 ′ ) 𝑒𝑒 i𝒒𝒒 ∙ 𝒓𝒓 ′ ℏ ⁄ d 𝒓𝒓 ′ ∞−∞ . (111) Here, 𝑓𝑓 ( 𝒒𝒒 ) has the dimensions of length (meter in 𝑆𝑆𝑆𝑆 ). Note that the presence of 𝑒𝑒 i𝑘𝑘𝑟𝑟 𝑟𝑟⁄ in Eq.(110) makes the wave-function 𝜓𝜓 ( 𝒓𝒓 ) dimensionless. Let d Ω = sin 𝜃𝜃 d 𝜃𝜃 d 𝜑𝜑 be the differential element of the solid angle viewed from the origin of the coordinates. The differential scattering cross-section will then be d 𝜎𝜎 d Ω⁄ = | 𝑓𝑓 ( 𝒒𝒒 )| , with the total cross-section being 𝜎𝜎 = ∫ | 𝑓𝑓 ( 𝒒𝒒 )| d Ω . d 𝒓𝒓 ′ stands for d 𝑥𝑥 ′ d 𝑥𝑥 ′ d 𝑧𝑧 ′ Example 1 . A particle of mass 𝓂𝓂 is scattered by the spherically symmetric potential 𝑉𝑉 ( 𝑟𝑟 ) corresponding to a fixed particle located at 𝒓𝒓 = 0 . The scattering amplitude, computed from Eq.(111), will be 𝑓𝑓 ( 𝒒𝒒 ) = − 𝓂𝓂2𝜋𝜋ℏ � ∫ 𝑉𝑉 ( 𝑟𝑟 ) exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ ) (2 𝜋𝜋𝑟𝑟 sin 𝜃𝜃 )d 𝜃𝜃 d 𝑟𝑟 𝜋𝜋𝜃𝜃=0∞𝑟𝑟=0 = − ∫ 𝑟𝑟𝑉𝑉 ( 𝑟𝑟 ) sin( 𝑞𝑞𝑟𝑟 ℏ⁄ ) d 𝑟𝑟 ∞0 .(112) Considering that 𝒒𝒒 = 𝒑𝒑 − 𝒑𝒑� , and that 𝒑𝒑 and 𝒑𝒑� have the same magnitude 𝑝𝑝 , we denote by 𝜃𝜃 the angle between 𝒑𝒑 and 𝒑𝒑� , and proceed to write 𝑞𝑞 = 2 𝑝𝑝 sin( 𝜃𝜃 ⁄ ) . The scattering amplitude thus has circular symmetry around the direction of the incident momentum 𝒑𝒑 . The ambiguity of Eq.(112) with regard to forward scattering at 𝜃𝜃 = 0 is resolved if the forward amplitude 𝑓𝑓 (0) is directly computed from Eq.(110) — with the destructive interference between the incident and scattered amplitudes properly taken into account. For the Yukawa potential 𝑉𝑉 ( 𝒓𝒓 ) = 𝑣𝑣 𝑒𝑒 −𝛼𝛼𝑟𝑟 𝑟𝑟⁄ , with 𝛼𝛼 > 0 being the range parameter, the scattering amplitude is obtained upon integrating Eq.(112), as follows: 𝑓𝑓 ( 𝒒𝒒 ) = − ℏ𝑞𝑞 ∫ 𝑒𝑒 −𝛼𝛼𝑟𝑟 sin( 𝑞𝑞𝑟𝑟 ℏ⁄ ) d 𝑟𝑟 ∞0 = − 𝑞𝑞 + ( 𝛼𝛼ℏ ) . (113) In the limit when 𝛼𝛼 → , the Yukawa potential approaches the Coulomb potential 𝑉𝑉 ( 𝒓𝒓 ) = 𝑣𝑣 𝑟𝑟⁄ . When a particle having electric charge ± 𝑒𝑒 and energy ℰ = 𝑝𝑝 𝓂𝓂⁄ is scattered from another particle of charge ± 𝑒𝑒 , the scattering cross-section will be d𝜎𝜎dΩ = | 𝑓𝑓 ( 𝒒𝒒 )| = � 𝑒𝑒 ℰ � ( 𝜃𝜃 2⁄ ) . (114) This is the famous Rutherford scattering cross-section. Example 2 . In low-energy scattering, 𝑘𝑘 ≅ and the scattering amplitude in all directions becomes 𝑓𝑓 ( 𝜃𝜃 , 𝜑𝜑 ) ≅ − ( 𝓂𝓂 𝜋𝜋ℏ ⁄ ) ∫ 𝑉𝑉 ( 𝒓𝒓 )d 𝒓𝒓 ∞−∞ . In the case of low-energy, soft-sphere scattering, where 𝑉𝑉 ( 𝒓𝒓 ) = 𝑉𝑉 when 𝑟𝑟 ≤ 𝑟𝑟 and zero otherwise, we find 𝑓𝑓 ( 𝜃𝜃 , 𝜑𝜑 ) = − ( 𝓂𝓂 𝜋𝜋ℏ ⁄ )(4 𝜋𝜋𝑟𝑟 ⁄ ) 𝑉𝑉 . (115a) d 𝜎𝜎 d Ω⁄ = | 𝑓𝑓 ( 𝜃𝜃 , 𝜑𝜑 )| = (2 𝓂𝓂𝑟𝑟 𝑉𝑉 ⁄ ) . (115b) 𝜎𝜎 = 4 𝜋𝜋 (d 𝜎𝜎 d Ω⁄ ) = 16 𝜋𝜋𝓂𝓂 𝑟𝑟 𝑉𝑉 ⁄ . (115c) Let us now consider the case of a polarized neutron entering a ferromagnetic medium and getting scattered from the host’s magnetic electrons. To obtain an estimate of the corresponding scattering potential 𝑉𝑉 ( 𝒓𝒓 ) , we begin by noting that the magnetic field surrounding a point-dipole 𝑙𝑙𝛿𝛿 ( 𝒓𝒓 ) 𝒛𝒛� in free space is 𝑯𝑯 ( 𝒓𝒓 ) = 𝑙𝑙 (2 cos 𝜃𝜃 𝒓𝒓� + sin 𝜃𝜃 𝜽𝜽� ) (4 𝜋𝜋𝜇𝜇 𝑟𝑟 ) ⁄ . (116) A second magnetic point-dipole 𝒎𝒎 ′ , located at 𝒓𝒓 ≠ , will have the energy ℰ = −𝒎𝒎 ′ ∙ 𝑯𝑯 ( 𝒓𝒓 ) , which may be written as follows: ℰ = −𝒎𝒎 ′ ∙ 𝑙𝑙 [3 cos 𝜃𝜃 𝒓𝒓� − (cos 𝜃𝜃 𝒓𝒓� − sin 𝜃𝜃 𝜽𝜽� )] (4 𝜋𝜋𝜇𝜇 𝑟𝑟 ) ⁄ = 𝒎𝒎 ′ ∙ 𝑙𝑙 ( 𝒛𝒛� − 𝜃𝜃 𝒓𝒓� ) (4 𝜋𝜋𝜇𝜇 𝑟𝑟 ) ⁄ = [ 𝒎𝒎 ∙ 𝒎𝒎 ′ − 𝒎𝒎 ∙ 𝒓𝒓� )( 𝒎𝒎 ′ ∙ 𝒓𝒓� )] (4 𝜋𝜋𝜇𝜇 𝑟𝑟 ) ⁄ . (117) These results are consistent with the Einstein-Laub formula 𝑭𝑭 = ( 𝒎𝒎 ∙ 𝜵𝜵 ) 𝑯𝑯 for the force as well as 𝑻𝑻 = 𝒎𝒎 × 𝑯𝑯 for the torque experienced by a point-dipole in a magnetic field. Recall
The magnetic dipole moment 𝑙𝑙 should not be confused with the particle’s mass 𝓂𝓂 . 𝑩𝑩 as 𝜇𝜇 𝑯𝑯 + 𝑴𝑴 maintains that the magnitude of 𝒎𝒎 equals 𝜇𝜇 times the electrical current times the area of a small current loop. Consequently, the aforementioned expression for ℰ coincides with the well-known expression ℰ = −𝒎𝒎 ∙ 𝑩𝑩 ( 𝒓𝒓 ) of the energy of the dipole 𝒎𝒎 in the external field 𝑩𝑩 ( 𝒓𝒓 ) = 𝜇𝜇 𝑯𝑯 ( 𝒓𝒓 ) . Equation (117) must be augmented by the contact term − 𝒎𝒎 ∙ 𝒎𝒎 ′ 𝛿𝛿 ( 𝒓𝒓 ) 3 𝜇𝜇 ⁄ to account for the energy of the dipole pair when 𝒎𝒎 and 𝒎𝒎 ′ overlap in space. We will have ℰ ( 𝒓𝒓 ) = 𝒎𝒎 ∙ 𝒎𝒎 ′ − ( 𝒎𝒎 ∙ 𝒓𝒓� )( 𝒎𝒎 ′ ∙ 𝒓𝒓� ) 𝑟𝑟 − ∙ 𝒎𝒎 ′ 𝛿𝛿 ( 𝒓𝒓 ) . (118) Suppose the electron has wave-function 𝜓𝜓 ( 𝒓𝒓 e ) and magnetic dipole moment 𝝁𝝁 e , while the incoming neutron has wave-function exp(i 𝒌𝒌 ∙ 𝒓𝒓 n ) , magnetic dipole moment 𝝁𝝁 n , and mass 𝓂𝓂 n . We assume the scattering process does not involve a spin flip, so that both 𝝁𝝁 e and 𝝁𝝁 n retain their orientations after the collision. Moreover, we assume the electron — being bound to its host lattice — does not get dislocated or otherwise distorted, so that 𝜓𝜓 ( 𝒓𝒓 e ) is the same before and after the collision. Thus, the potential energy distribution across the landscape of the incoming neutron is the integral over 𝒓𝒓 e of the product of the electron’s probability-density function | 𝜓𝜓 ( 𝒓𝒓 e )| and the dipole-dipole interaction energy ℰ ( 𝒓𝒓 e − 𝒓𝒓 n ) between the neutron and the electron. Consequently, in the first Born approximation, the scattering amplitude from an initial neutron momentum 𝒑𝒑 = ℏ𝒌𝒌 to a final momentum 𝒑𝒑 ′ = ℏ𝒌𝒌 ′ , is given by 𝑓𝑓 ( 𝒑𝒑 ′ , 𝒑𝒑 ) = −� 𝓂𝓂 n � ∫ | 𝜓𝜓 ( 𝒓𝒓 e )| ℰ ( 𝒓𝒓 n − 𝒓𝒓 e ) exp[i( 𝒌𝒌 − 𝒌𝒌 ′ ) ∙ 𝒓𝒓 n ] d 𝒓𝒓 e d 𝒓𝒓 n ∞−∞ . (119) Defining the electronic magnetization (i.e., magnetic moment density of the electron) by 𝑴𝑴 ( 𝒓𝒓 e ) = | 𝜓𝜓 ( 𝒓𝒓 e )| 𝝁𝝁 e and its Fourier transform by 𝑴𝑴� ( 𝒌𝒌 ) = ∫ 𝑴𝑴 ( 𝒓𝒓 e ) exp(i 𝒌𝒌 ∙ 𝒓𝒓 e ) d 𝒓𝒓 e∞−∞ , upon substitution from Eq.(118) into Eq.(119) and setting 𝒓𝒓 = 𝒓𝒓 n – 𝒓𝒓 e , we find 𝑓𝑓 ( 𝒑𝒑 ′ , 𝒑𝒑 ) = − 𝓂𝓂 n 𝜇𝜇 ℏ � � 𝝁𝝁 n −3 ( 𝝁𝝁 n ∙ 𝒓𝒓� ) 𝒓𝒓�𝑟𝑟 – ( 𝒓𝒓 ) 𝝁𝝁 n � ∙ 𝑴𝑴 ( 𝒓𝒓 e ) exp[i( 𝒑𝒑 − 𝒑𝒑 ′ ) ∙ 𝒓𝒓 n ℏ⁄ ] d 𝒓𝒓 e d 𝒓𝒓 n ∞−∞ . (120) Defining 𝒒𝒒 = 𝒑𝒑 − 𝒑𝒑 ′ and changing the variables from ( 𝒓𝒓 e , 𝒓𝒓 n ) to ( 𝒓𝒓 e , 𝒓𝒓 = 𝒓𝒓 n − 𝒓𝒓 e ) — whose transformation Jacobian is — substantially simplifies the above integral, yielding 𝑓𝑓 ( 𝒒𝒒 ) = − 𝓂𝓂 n 𝜇𝜇 ℏ � � 𝝁𝝁 n −3 ( 𝝁𝝁 n ∙ 𝒓𝒓� ) 𝒓𝒓�𝑟𝑟 – ( 𝒓𝒓 ) 𝝁𝝁 n � ∙ 𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) exp(i 𝒒𝒒 ∙ 𝒓𝒓 ℏ⁄ ) d 𝒓𝒓 ∞−∞ . (121) Appendix I shows that the exact evaluation of the integral in Eq.(121) leads to 𝑓𝑓 ( 𝒒𝒒 ) = � 𝓂𝓂 n ℏ � 𝝁𝝁 n ∙ �𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) − [ 𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) ∙ 𝒒𝒒� ] 𝒒𝒒�� . (122) This is the same result as given in Ref. [29], Eq.(23), in the case of 𝜆𝜆 = 1 . The coefficient 𝜋𝜋𝜇𝜇 appears here because we have worked in the 𝑆𝑆𝑆𝑆 system of units with 𝑩𝑩 = 𝜇𝜇 𝑯𝑯 + 𝑴𝑴 .
11. Concluding remarks . In this paper, we described some of the fundamental theories of EM scattering and diffraction using the electrodynamics of Maxwell and Lorentz in conjunction with standard mathematical methods of the vector calculus, complex analysis, differential equations, and Fourier transform theory. The scalar Huygens-Fresnel-Kirchhoff and Rayleigh-Sommerfeld theories were presented at first, followed by their extensions that cover the case of vector diffraction of EM waves. Examples were provided to showcase the application of these vector diffraction and scattering formulas to certain problems of practical interest. We did not discuss 7 the alternate method of diffraction calculations by means of Fourier transformation, which involves an expansion of the initial field profile in the 𝑥𝑥𝑥𝑥 -plane at 𝑧𝑧 = 0 into its plane-wave constituents. In fact, with the aid of the two-dimensional Fourier transform of 𝐺𝐺 ( 𝒓𝒓 , 0) given in Eq.(37), it is rather easy to establish the equivalence of the Fourier expansion method with the Rayleigh-Sommerfeld formula in Eq.(25), and also with the related vector formulas in Eqs.(32) and (33). Appendix J outlines the mathematical steps needed to establish these equivalencies. The Sommerfeld solution to the problem of diffraction from a thin, perfectly conducting half-plane described in Sec.7 is one of the few problems in the EM theory of diffraction for which an exact analytical solution has been found; for a discussion of related problems of this type, see Ref.[1], Chapter 11. Scattering of plane-waves from spherical particles of known relative permittivity 𝜀𝜀 ( 𝜔𝜔 ) and permeability 𝜇𝜇 ( 𝜔𝜔 ) , the so-called Mie scattering, is another problem for which an exact solution (albeit in the form of an infinite series) exists; for a discussion of this and related problems the reader is referred to the vast literature of Mie scattering. In our analysis of neutron scattering from ferromagnets in Sec.10, we used the contact term − 𝒎𝒎 ∙ 𝒎𝒎 ′ 𝛿𝛿 ( 𝒓𝒓 ) 3 𝜇𝜇 ⁄ to account for the interaction energy of the dipole pair 𝒎𝒎 , 𝒎𝒎 ′ when they happen to overlap at the same location in space. This is tantamount to assuming that the dipolar magnetic moments are produced by circulating electrical currents. The contact term would have been 𝒎𝒎 ∙ 𝒎𝒎 ′ 𝛿𝛿 ( 𝒓𝒓 ) 3 𝜇𝜇 ⁄ had each magnetic moment been produced by a pair of equal and opposite magnetic monopoles residing within the corresponding particle. Since the Amperian current loop model has been found to agree most closely with experimental findings, we used the former expression for the contact term in Eq.(118); see Ref. [9], Sec.5.7, and Ref. [29] for a pedagogical discussion of the experimental evidence — from neutron scattering as well as the existence of the famous
21 cm astrophysical spectral line of atomic hydrogen — in favor of the Amperian current loop model of the intrinsic magnetic dipole moments of subatomic particles. Although we did not discuss the Babinet principle of complementary screens that is well known in classical optics, it is worth mentioning here that a rigorous version of this principle has been proven in Maxwellian electrodynamics.
The original version of Babinet’s principle is based on the Kirchhoff diffraction integral of Eq.(23), and the notion that, if 𝑆𝑆 consists of apertures in an opaque screen, then the complement of 𝑆𝑆 would be opaque where 𝑆𝑆 is transmissive, and transmissive where 𝑆𝑆 is opaque. Considering that, in Kirchhoff’s approximation, 𝜓𝜓 ( 𝒓𝒓 ) and 𝜕𝜕 𝑛𝑛 𝜓𝜓 ( 𝒓𝒓 ) in Eq.(23) retain the values of the incident beam in the open aperture(s) but vanish in the opaque regions, it is a reasonable conjecture that the observed field in the presence of 𝑆𝑆 and that in the presence of 𝑆𝑆 ’s complement would add up to the observed field when all screens are removed — i.e., when the unobstructed beam reaches the observation point. Similar arguments can be based on either of the Rayleigh-Sommerfeld diffraction integrals in Eqs.(24) and (25), provided, of course, that the Kirchhoff approximation remains applicable. Appendix K describes the rigorous version of the Babinet principle and provides a simple proof that relies on symmetry arguments similar to those used in Example 2 of Sec.6. Finally, to keep the size and scope of this tutorial within reasonable boundaries, we did not broach the important problem of EM scattering from small dielectric spheres, nor that of EM scattering from small perfectly conducting spheres. The interested reader can find a detailed discussion of these problems in Appendices L and M, respectively. 8 References
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Applied Physics A , pp1-11 (2017). Appendix A
The integrals in Eqs.(17) and (18) are evaluated with the aid of Euler’s beta and gamma functions, 𝐵𝐵 ( 𝑥𝑥 , 𝑦𝑦 ) and Γ ( 𝑥𝑥 ) , as follows: ∫ 𝜋𝜋𝑟𝑟 ( 𝑟𝑟 + 𝜀𝜀 ) −5 2⁄ d 𝑟𝑟 ∞0 = 4 𝜋𝜋𝜀𝜀 −5 2⁄ ∫ 𝑟𝑟 (1 + 𝜀𝜀 −1 𝑟𝑟 ) −5 2⁄ d 𝑟𝑟 ∞0 = 2 𝜋𝜋𝜀𝜀 −5 2⁄ 𝜀𝜀 𝐵𝐵 � , 1 � = 2 𝜋𝜋𝜀𝜀 −1 Γ ( ) Γ ( ) Γ ( ) = 4 𝜋𝜋 𝜀𝜀⁄ . (A1) ∫ 𝜋𝜋𝑟𝑟 ( 𝑟𝑟 + 𝜀𝜀 ) −3 2⁄ d 𝑟𝑟 ∞0 = 4 𝜋𝜋𝜀𝜀 −3 2⁄ ∫ 𝑟𝑟 (1 + 𝜀𝜀 −1 𝑟𝑟 ) −3 2⁄ d 𝑟𝑟 ∞0 = 2 𝜋𝜋𝜀𝜀 −3 2⁄ 𝜀𝜀𝐵𝐵 � � = 2 𝜋𝜋𝜀𝜀 − ½ Γ ( ) Γ (½) Γ ( ) = 4 𝜋𝜋 √𝜀𝜀⁄ . (A2) In these derivations, the following identities from Gradshteyn and Ryzhik’s Table of Integrals, Series, and Products have been used: � 𝑥𝑥 𝜇𝜇−1 (1 + 𝛽𝛽𝑥𝑥 𝑝𝑝 ) −𝜈𝜈 d 𝑥𝑥 = 𝑝𝑝 −1 𝛽𝛽 −𝜇𝜇 𝑝𝑝⁄ 𝐵𝐵 � 𝜇𝜇𝑝𝑝 , 𝜈𝜈 − 𝜇𝜇𝑝𝑝 � ∞0 ; |arg( 𝛽𝛽 )| < 𝜋𝜋 , 𝑝𝑝 > 0, 0 < Re( 𝜇𝜇 ) < 𝑝𝑝 Re( 𝜈𝜈 ) . (G&R -11) 𝐵𝐵 ( 𝑥𝑥 , 𝑦𝑦 ) = Γ ( 𝑥𝑥 ) Γ ( 𝑦𝑦 ) Γ ( 𝑥𝑥 + 𝑦𝑦 ) ⁄ . (G&R -1) Γ ( 𝑥𝑥 + 1) = 𝑥𝑥Γ ( 𝑥𝑥 ) . (G&R -1) Γ (1) = Γ (2) = 1; Γ (½) = √𝜋𝜋 . (G&R -1,2) Appendix B
The Green function 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) is the solution of the inhomogeneous Helmholtz equation: ( 𝛻𝛻 + 𝑘𝑘 ) 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) = − 𝜋𝜋𝜋𝜋 ( 𝒓𝒓 − 𝒓𝒓 ) . (B1) Since the function 𝐺𝐺 is expected to be translation invariant, we solve Eq.(B1) for 𝒓𝒓 = 0 . The function of interest here, namely, the Fourier transform 𝐺𝐺� ( 𝒌𝒌 ) of 𝐺𝐺 ( 𝒓𝒓 , 0) , is thus seen to be 𝐺𝐺� ( 𝒌𝒌 ) = 4 𝜋𝜋 ( 𝑘𝑘 − 𝑘𝑘 ) ⁄ . (B2) The inverse Fourier transform of 𝐺𝐺� ( 𝒌𝒌 ) can now be found by direct integration, as follows: 𝐺𝐺 ( 𝒓𝒓 , 0) = (2 𝜋𝜋 ) −3 ∭ 𝐺𝐺� ( 𝒌𝒌 ) 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 d 𝒌𝒌 ∞−∞ = (2 𝜋𝜋 ) −3 � � − 𝑘𝑘 𝑒𝑒 i𝑘𝑘𝑘𝑘 cos 𝜃𝜃 𝜋𝜋𝑘𝑘 sin 𝜃𝜃 d 𝑘𝑘 d 𝜃𝜃 𝜋𝜋𝜃𝜃=0∞𝑘𝑘=0 = i𝜋𝜋𝑘𝑘 � 𝑘𝑘𝑘𝑘 − 𝑘𝑘 �𝑒𝑒 i𝑘𝑘𝑘𝑘 cos 𝜃𝜃 � 𝜃𝜃=0𝜋𝜋 � d 𝑘𝑘 ∞𝑘𝑘=0 = i𝜋𝜋𝑘𝑘 � 𝑘𝑘𝑘𝑘 − 𝑘𝑘 �𝑒𝑒 −i𝑘𝑘𝑘𝑘 − 𝑒𝑒 i𝑘𝑘𝑘𝑘 � d 𝑘𝑘 ∞0 = i2𝜋𝜋𝑘𝑘 �� 𝑘𝑘 exp ( −i𝑘𝑘𝑘𝑘 ) 𝑘𝑘 − 𝑘𝑘 d 𝑘𝑘 ∞−∞ − � 𝑘𝑘 exp ( i𝑘𝑘𝑘𝑘 ) 𝑘𝑘 − 𝑘𝑘 d 𝑘𝑘 ∞−∞ � = ½ � 𝑒𝑒 i𝑘𝑘0𝑟𝑟 𝑘𝑘 + 𝑒𝑒 −i𝑘𝑘0𝑟𝑟 𝑘𝑘 � . (B3) It is noteworthy that 𝐺𝐺 ( 𝒓𝒓 , 0) has equal contributions from exp(±i 𝑘𝑘 𝑟𝑟 ) 𝑟𝑟⁄ , even though, in practice, one typically picks only one of these two solutions — either the outgoing spherical wave with the plus sign or the incoming wave with the minus sign. The solution of Eq.(B1), of course, is not unique, since any plane-wave in the form of 𝐴𝐴 exp(i 𝑘𝑘 𝝈𝝈 ∙ 𝒓𝒓 ) , where 𝐴𝐴 is an residue calculus d 𝒌𝒌 stands for d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦 d 𝑘𝑘 𝑧𝑧 𝝈𝝈 an arbitrary (real-valued) unit-vector, can be added to 𝐺𝐺 ( 𝒓𝒓 , 0) without affecting the solution. A superposition of such plane-waves can shift the balance between exp(±i 𝑘𝑘 𝑟𝑟 ) 𝑟𝑟⁄ in Eq.(B3), so that any linear combination of these two functions can act as a perfectly good solution of Eq.(B1). As a matter of fact, sin( 𝑘𝑘 𝑟𝑟 ) 𝑟𝑟⁄ is one such homogeneous solution of Eq.(B1), whose linear combination with Eq.(B3) can produce the desired balance. It is possible to directly compute the Fourier transform of either of the candidate solutions for 𝐺𝐺 ( 𝒓𝒓 , 0) , namely, exp(±i 𝑘𝑘 𝑟𝑟 ) 𝑟𝑟⁄ . We begin by replacing 𝑘𝑘 with 𝑘𝑘 − i 𝜀𝜀 , where 𝜀𝜀 , a small (positive or negative) number, will eventually be made to approach zero. We find ℱ � exp [± i ( 𝑘𝑘 −i𝜀𝜀 ) 𝑘𝑘 ] 𝑘𝑘 � = � � exp [± i ( 𝑘𝑘 −i𝜀𝜀 ) 𝑘𝑘 ] 𝑘𝑘 𝑒𝑒 −i𝑘𝑘𝑘𝑘 cos 𝜃𝜃 𝜋𝜋𝑟𝑟 sin 𝜃𝜃 d 𝑟𝑟 d 𝜃𝜃 𝜋𝜋𝜃𝜃=0∞𝑟𝑟=0 = � 𝑒𝑒 ± i ( 𝑘𝑘 −i𝜀𝜀 ) 𝑘𝑘 �𝑒𝑒 −i𝑘𝑘𝑘𝑘 cos 𝜃𝜃 � 𝜃𝜃=0𝜋𝜋 � d 𝑟𝑟 ∞𝑟𝑟=0 = � 𝑒𝑒 ± i ( 𝑘𝑘 −i𝜀𝜀 ) 𝑘𝑘 �𝑒𝑒 i𝑘𝑘𝑘𝑘 − 𝑒𝑒 −i𝑘𝑘𝑘𝑘 � d 𝑟𝑟 ∞𝑟𝑟=0 = ∫ 𝑒𝑒 ± 𝜀𝜀𝑘𝑘 {exp[i(± 𝑘𝑘 + 𝑘𝑘 ) 𝑟𝑟 ] − exp[i(± 𝑘𝑘 − 𝑘𝑘 ) 𝑟𝑟 ]}d 𝑟𝑟 ∞0 = �− ± 𝜀𝜀+i (± 𝑘𝑘 +𝑘𝑘 ) + ± 𝜀𝜀+i (± 𝑘𝑘 −𝑘𝑘 ) � = � ±( 𝑘𝑘 −i𝜀𝜀 ) + 𝑘𝑘 − ±( 𝑘𝑘 −i𝜀𝜀 ) − 𝑘𝑘 � = − ( 𝑘𝑘 −i𝜀𝜀 ) . (B4) It is thus seen that 𝐺𝐺� ( 𝒌𝒌 ) = 4 𝜋𝜋 ( 𝑘𝑘 − 𝑘𝑘 ) ⁄ is the Fourier transform of 𝐺𝐺 ( 𝒓𝒓 , 0) = 𝑒𝑒 ± i𝑘𝑘 𝑘𝑘 𝑟𝑟⁄ , irrespective of the sign of 𝑘𝑘 . Appendix C
Kirchhoff’s Eq.(22), applied to the 𝑥𝑥 -component of the 𝐸𝐸 -field yields Eq.(27), which should similarly be satisfied by the remaining components 𝐸𝐸 𝑦𝑦 , 𝐸𝐸 𝑧𝑧 of the 𝐸𝐸 -field. The vectorial version of the Kirchhoff formula may now be written down for 𝑬𝑬 ( 𝒓𝒓 ) , which, upon manipulation by standard vector identities, leads to Eq.(28). A step-by-step procedure is shown below. 𝜋𝜋𝑬𝑬 ( 𝒓𝒓 ) = 2 ∫ ( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝑬𝑬 d 𝑠𝑠 𝑆𝑆 + ∫ 𝜵𝜵 ( 𝐺𝐺𝑬𝑬 )d 𝒓𝒓 𝑉𝑉 = 2 ∫ ( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝑬𝑬 d 𝑠𝑠 𝑆𝑆 + ∫ 𝜵𝜵 [ 𝜵𝜵 ∙ ( 𝐺𝐺𝑬𝑬 )]d 𝒓𝒓 𝑉𝑉 − ∫ 𝜵𝜵 × [ 𝜵𝜵 × ( 𝐺𝐺𝑬𝑬 )]d 𝒓𝒓 𝑉𝑉 = 2 ∫ ( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝑬𝑬 d 𝑠𝑠 𝑆𝑆 − ∫ [ 𝜵𝜵 ∙ ( 𝐺𝐺𝑬𝑬 )] 𝒏𝒏� d 𝑠𝑠 𝑆𝑆 + ∫ 𝒏𝒏� × [ 𝜵𝜵 × ( 𝐺𝐺𝑬𝑬 )]d 𝑠𝑠 𝑆𝑆 = ∫ [2( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝑬𝑬 − ( 𝐺𝐺𝜵𝜵 ∙ 𝑬𝑬 + 𝑬𝑬 ∙ 𝜵𝜵𝐺𝐺 ) 𝒏𝒏� + 𝒏𝒏� × ( 𝜵𝜵𝐺𝐺 × 𝑬𝑬 + 𝐺𝐺𝜵𝜵 × 𝑬𝑬 )]d 𝑠𝑠 𝑆𝑆 = ∫ [( 𝒏𝒏� ∙ 𝜵𝜵𝐺𝐺 ) 𝑬𝑬 − ( 𝑬𝑬 ∙ 𝜵𝜵𝐺𝐺 ) 𝒏𝒏� + ( 𝒏𝒏� ∙ 𝑬𝑬 ) 𝜵𝜵𝐺𝐺 + i 𝜔𝜔 ( 𝒏𝒏� × 𝑩𝑩 ) 𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆 = ∫ [( 𝒏𝒏� × 𝑬𝑬 ) × 𝜵𝜵𝐺𝐺 + ( 𝒏𝒏� ∙ 𝑬𝑬 ) 𝜵𝜵𝐺𝐺 + i 𝜔𝜔 ( 𝒏𝒏� × 𝑩𝑩 ) 𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆 . (C1) We have thus arrived at a vectorial version of the Kirchhoff formula for the observed 𝐸𝐸 -field. i 𝜔𝜔𝑩𝑩 ∫ 𝜵𝜵𝜙𝜙 ( 𝒓𝒓 )d 𝒓𝒓 = − 𝑉𝑉 ∫ 𝜙𝜙 ( 𝒓𝒓 ) 𝒏𝒏� d 𝑠𝑠 𝑆𝑆 𝜵𝜵 𝑽𝑽 ( 𝒓𝒓 ) = ( 𝛻𝛻 𝑉𝑉 𝑥𝑥 ) 𝒙𝒙� + ( 𝛻𝛻 𝑉𝑉 𝑦𝑦 ) 𝒚𝒚� + ( 𝛻𝛻 𝑉𝑉 𝑧𝑧 ) 𝒛𝒛� 𝜵𝜵 𝑽𝑽 = 𝜵𝜵 ( 𝜵𝜵 ∙ 𝑽𝑽 ) − 𝜵𝜵 × ( 𝜵𝜵 × 𝑽𝑽 ) 𝒏𝒏� points into the volume 𝑉𝑉 ∫ 𝜵𝜵 × 𝑽𝑽 ( 𝒓𝒓 )d 𝒓𝒓 = − 𝑉𝑉 ∫ 𝒏𝒏� × 𝑽𝑽 ( 𝒓𝒓 )d 𝑠𝑠 𝑆𝑆 𝜵𝜵 ∙ ( 𝜙𝜙𝑽𝑽 ) = 𝜙𝜙𝜵𝜵 ∙ 𝑽𝑽 + 𝑽𝑽 ∙ 𝜵𝜵𝜙𝜙 𝜵𝜵 × ( 𝜙𝜙𝑽𝑽 ) = 𝜵𝜵𝜙𝜙 × 𝑽𝑽 + 𝜙𝜙𝜵𝜵 × 𝑽𝑽 𝒂𝒂 × ( 𝒃𝒃 × 𝒄𝒄 ) = ( 𝒂𝒂 ∙ 𝒄𝒄 ) 𝒃𝒃 − ( 𝒂𝒂 ∙ 𝒃𝒃 ) 𝒄𝒄 Maxwell’s 1 st equation Fourier operator Choose sign of 𝜀𝜀 so that exp(± 𝜀𝜀𝑟𝑟 ) → when 𝑟𝑟 → ∞ . Appendix D
It is possible to derive Eq.(30) directly from Eq.(29), and vice versa. Here, Maxwell’s equation 𝜵𝜵 × 𝑩𝑩 ( 𝒓𝒓 ) = − (i 𝜔𝜔 𝑐𝑐 ⁄ ) 𝑬𝑬 ( 𝒓𝒓 ) will be used to show that Eq.(29) is a direct consequence of Eq.(30). Since the integration variable over the closed surface 𝑆𝑆 is 𝒓𝒓 , whereas the fields are observed at 𝒓𝒓 , we denote the differential operator by 𝜵𝜵 when operating on 𝒓𝒓 , and by 𝜵𝜵 when operating on 𝒓𝒓 . The symmetry of 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 ) then allows us to replace 𝜵𝜵 𝐺𝐺 with −𝜵𝜵𝐺𝐺 . Starting with 𝑩𝑩 ( 𝒓𝒓 ) given in Eq.(30) as an integral over the closed surface 𝑆𝑆 , we will have 𝑬𝑬 ( 𝒓𝒓 ) = (i 𝑐𝑐 𝜔𝜔⁄ ) 𝜵𝜵 × 𝑩𝑩 ( 𝒓𝒓 ) = (i 𝑐𝑐 𝜋𝜋𝜔𝜔⁄ ) ∮ 𝜵𝜵 × [( 𝒏𝒏� × 𝑩𝑩 ) × 𝜵𝜵𝐺𝐺 + ( 𝒏𝒏� ∙ 𝑩𝑩 ) 𝜵𝜵𝐺𝐺 − i( 𝜔𝜔 𝑐𝑐 ⁄ )( 𝒏𝒏� × 𝑬𝑬 ) 𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆 = (i 𝑐𝑐 𝜋𝜋𝜔𝜔⁄ ) ∮ {( 𝒏𝒏� × 𝑩𝑩 )( 𝜵𝜵 ∙ 𝜵𝜵𝐺𝐺 ) − [( 𝒏𝒏� × 𝑩𝑩 ) ∙ 𝜵𝜵 ] 𝜵𝜵𝐺𝐺 + ( 𝒏𝒏� ∙ 𝑩𝑩 )( 𝜵𝜵 × 𝜵𝜵𝐺𝐺 ) 𝑆𝑆 − i( 𝜔𝜔 𝑐𝑐 ⁄ )( 𝜵𝜵 𝐺𝐺 ) × ( 𝒏𝒏� × 𝑬𝑬 )}d 𝑠𝑠 = (i 𝑐𝑐 𝜋𝜋𝜔𝜔⁄ ) ∮ {( 𝒏𝒏� × 𝑩𝑩 )[4 𝜋𝜋𝜋𝜋 ( 𝒓𝒓 − 𝒓𝒓 ) + 𝑘𝑘 𝐺𝐺 ] + [( 𝒏𝒏� × 𝑩𝑩 ) ∙ 𝜵𝜵 ] 𝜵𝜵𝐺𝐺 𝑆𝑆 − i( 𝜔𝜔 𝑐𝑐 ⁄ )( 𝒏𝒏� × 𝑬𝑬 ) × 𝜵𝜵𝐺𝐺 }d 𝑠𝑠 = (4 𝜋𝜋 ) −1 ∮ {i 𝜔𝜔 ( 𝒏𝒏� × 𝑩𝑩 ) 𝐺𝐺 + (i 𝑐𝑐 𝜔𝜔⁄ )[( 𝒏𝒏� × 𝑩𝑩 ) ∙ 𝜵𝜵 ] 𝜵𝜵𝐺𝐺 + ( 𝒏𝒏� × 𝑬𝑬 ) × 𝜵𝜵𝐺𝐺 }d 𝑠𝑠 𝑆𝑆 . (D1) The middle term in the above integrand can be expanded, as follows: [( 𝒏𝒏� × 𝑩𝑩 ) ∙ 𝜵𝜵 ] 𝜵𝜵𝐺𝐺 = [( 𝑛𝑛 𝑦𝑦 𝐵𝐵 𝑧𝑧 − 𝑛𝑛 𝑧𝑧 𝐵𝐵 𝑦𝑦 ) 𝜕𝜕 𝑥𝑥 + ( 𝑛𝑛 𝑧𝑧 𝐵𝐵 𝑥𝑥 − 𝑛𝑛 𝑥𝑥 𝐵𝐵 𝑧𝑧 ) 𝜕𝜕 𝑦𝑦 + ( 𝑛𝑛 𝑥𝑥 𝐵𝐵 𝑦𝑦 − 𝑛𝑛 𝑦𝑦 𝐵𝐵 𝑥𝑥 ) 𝜕𝜕 𝑧𝑧 ] 𝜵𝜵𝐺𝐺 = [ 𝑛𝑛 𝑥𝑥 ( 𝐵𝐵 𝑦𝑦 𝜕𝜕 𝑧𝑧 − 𝐵𝐵 𝑧𝑧 𝜕𝜕 𝑦𝑦 ) + 𝑛𝑛 𝑦𝑦 ( 𝐵𝐵 𝑧𝑧 𝜕𝜕 𝑥𝑥 − 𝐵𝐵 𝑥𝑥 𝜕𝜕 𝑧𝑧 ) + 𝑛𝑛 𝑧𝑧 ( 𝐵𝐵 𝑥𝑥 𝜕𝜕 𝑦𝑦 − 𝐵𝐵 𝑦𝑦 𝜕𝜕 𝑥𝑥 )] 𝜵𝜵𝐺𝐺 = 𝑛𝑛 𝑥𝑥 [ 𝜕𝜕 𝑧𝑧 ( 𝐵𝐵 𝑦𝑦 𝜵𝜵𝐺𝐺 ) − 𝜕𝜕 𝑦𝑦 ( 𝐵𝐵 𝑧𝑧 𝜵𝜵𝐺𝐺 ) + ( 𝜕𝜕 𝑦𝑦 𝐵𝐵 𝑧𝑧 − 𝜕𝜕 𝑧𝑧 𝐵𝐵 𝑦𝑦 ) 𝜵𝜵𝐺𝐺 ] + 𝑛𝑛 𝑦𝑦 [ 𝜕𝜕 𝑥𝑥 ( 𝐵𝐵 𝑧𝑧 𝜵𝜵𝐺𝐺 ) − 𝜕𝜕 𝑧𝑧 ( 𝐵𝐵 𝑥𝑥 𝜵𝜵𝐺𝐺 ) + ( 𝜕𝜕 𝑧𝑧 𝐵𝐵 𝑥𝑥 − 𝜕𝜕 𝑥𝑥 𝐵𝐵 𝑧𝑧 ) 𝜵𝜵𝐺𝐺 ] + 𝑛𝑛 𝑧𝑧 [ 𝜕𝜕 𝑦𝑦 ( 𝐵𝐵 𝑥𝑥 𝜵𝜵𝐺𝐺 ) − 𝜕𝜕 𝑥𝑥 ( 𝐵𝐵 𝑦𝑦 𝜵𝜵𝐺𝐺 ) + ( 𝜕𝜕 𝑥𝑥 𝐵𝐵 𝑦𝑦 − 𝜕𝜕 𝑦𝑦 𝐵𝐵 𝑥𝑥 ) 𝜵𝜵𝐺𝐺 ] = [ 𝒏𝒏� ∙ 𝜵𝜵 × ( 𝑩𝑩𝜕𝜕 𝑥𝑥 𝐺𝐺 )] 𝒙𝒙� + [ 𝒏𝒏� ∙ 𝜵𝜵 × ( 𝑩𝑩𝜕𝜕 𝑦𝑦 𝐺𝐺 )] 𝒚𝒚� + [ 𝒏𝒏� ∙ 𝜵𝜵 × ( 𝑩𝑩𝜕𝜕 𝑧𝑧 𝐺𝐺 )] 𝒛𝒛� +( 𝒏𝒏� ∙ 𝜵𝜵 × 𝑩𝑩 ) 𝜵𝜵𝐺𝐺 . (D2) Gauss’s theorem now allows us to replace the surface integral of 𝒏𝒏� ∙ 𝜵𝜵 × ( 𝑩𝑩𝜕𝜕 𝑥𝑥 𝐺𝐺 ) with the volume integral of 𝜵𝜵 ∙ [ 𝜵𝜵 × ( 𝑩𝑩𝜕𝜕 𝑥𝑥 𝐺𝐺 )] , which is zero since the divergence of the curl of any vector field is identically zero. By the same token, the surface integrals of the 2 nd and 3 rd terms on the right-hand side of Eq.(D2) vanish. Substituting the remaining term into Eq.(D1) and recalling that (i 𝑐𝑐 𝜔𝜔⁄ ) 𝜵𝜵 × 𝑩𝑩 = 𝑬𝑬 , we arrive at 𝑬𝑬 ( 𝒓𝒓 ) = (4 𝜋𝜋 ) −1 ∮ [i 𝜔𝜔 ( 𝒏𝒏� × 𝑩𝑩 ) 𝐺𝐺 + ( 𝒏𝒏� ∙ 𝑬𝑬 ) 𝜵𝜵𝐺𝐺 + ( 𝒏𝒏� × 𝑬𝑬 ) × 𝜵𝜵𝐺𝐺 ]d 𝑠𝑠 𝑆𝑆 . (D3) This is the same expression for 𝑬𝑬 ( 𝒓𝒓 ) as that in Eq.(28). The proof will be complete when the contribution of the remote spherical (or hemi-spherical) surface 𝑆𝑆 to the overall integral is recognized as negligible. Consequently, the integral over the closed surface 𝑆𝑆 = 𝑆𝑆 + 𝑆𝑆 in Eq.(D3) is equivalent to the integral over 𝑆𝑆 in Eq.(29). −𝜵𝜵 × 𝜵𝜵𝐺𝐺 = 0 𝜵𝜵 × ( 𝒂𝒂 × 𝒃𝒃 ) = 𝒂𝒂 ( 𝜵𝜵 ∙ 𝒃𝒃 ) − 𝒃𝒃 ( 𝜵𝜵 ∙ 𝒂𝒂 ) + ( 𝒃𝒃 ∙ 𝜵𝜵 ) 𝒂𝒂 − ( 𝒂𝒂 ∙ 𝜵𝜵 ) 𝒃𝒃 𝜵𝜵 × ( 𝜓𝜓𝒂𝒂 ) = 𝜵𝜵𝜓𝜓 × 𝒂𝒂 + 𝜓𝜓𝜵𝜵 × 𝒂𝒂 −𝜵𝜵 𝐺𝐺 𝜋𝜋 ( 𝒓𝒓 − 𝒓𝒓 ) = 0 on the surface 𝑆𝑆 ; 𝑘𝑘 = 𝜔𝜔 𝑐𝑐⁄ Appendix E
Evaluating the 2D Fourier transform relation of Eq.(37) requires the following identities: � sin�𝑎𝑎√𝑥𝑥 +𝑏𝑏 � cos ( 𝑐𝑐𝑥𝑥 ) √𝑥𝑥 +𝑏𝑏 d 𝑥𝑥 ∞0 = � ½ 𝜋𝜋𝐽𝐽 �𝑏𝑏√𝑎𝑎 − 𝑐𝑐 � , 0 < 𝑐𝑐 < 𝑎𝑎
0, 0 < 𝑎𝑎 < 𝑐𝑐 ( 𝑏𝑏 > 0) . (G&R � cos�𝑎𝑎√𝑥𝑥 +𝑏𝑏 � cos ( 𝑐𝑐𝑥𝑥 ) √𝑥𝑥 +𝑏𝑏 d 𝑥𝑥 ∞0 = �− ½ 𝜋𝜋𝑌𝑌 �𝑏𝑏√𝑎𝑎 − 𝑐𝑐 � , 0 < 𝑐𝑐 < 𝑎𝑎𝐾𝐾 �𝑏𝑏√𝑐𝑐 − 𝑎𝑎 � , 0 < 𝑎𝑎 < 𝑐𝑐 ( 𝑏𝑏 > 0) . (G&R � 𝐽𝐽 �𝛼𝛼√𝑥𝑥 + 𝑧𝑧 � cos( 𝛽𝛽𝑥𝑥 ) d 𝑥𝑥 ∞0 = � cos�𝑧𝑧�𝛼𝛼 −𝛽𝛽 ��𝛼𝛼 −𝛽𝛽 , 0 < 𝛽𝛽 < 𝛼𝛼
0, 0 < 𝛼𝛼 < 𝛽𝛽 ( 𝑧𝑧 > 0) . (G&R � 𝑌𝑌 �𝛼𝛼√𝑥𝑥 + 𝑧𝑧 � cos( 𝛽𝛽𝑥𝑥 ) d 𝑥𝑥 ∞0 = ⎩⎨⎧ sin�𝑧𝑧�𝛼𝛼 −𝛽𝛽 ��𝛼𝛼 −𝛽𝛽 , 0 < 𝛽𝛽 < 𝛼𝛼− exp�−𝑧𝑧�𝛽𝛽 −𝛼𝛼 ��𝛽𝛽 −𝛼𝛼 , 0 < 𝛼𝛼 < 𝛽𝛽 ( 𝑧𝑧 > 0) . (G&R � 𝐾𝐾 �𝛼𝛼√𝑥𝑥 + 𝑧𝑧 � cos( 𝛽𝛽𝑥𝑥 ) d 𝑥𝑥 ∞0 = 𝜋𝜋 exp�−𝑧𝑧�𝛼𝛼 +𝛽𝛽 �2�𝛼𝛼 +𝛽𝛽 , [ 𝑧𝑧 > 0, Re( 𝛼𝛼 ) > 0, 𝛽𝛽 > 0] . (G&R Here, 𝐽𝐽 ( ∙ ) is the Bessel function of first kind, order zero, 𝑌𝑌 ( ∙ ) is the Bessel function of second kind, order zero, and 𝐾𝐾 ( ∙ ) is the modified Bessel function of imaginary argument, order zero, defined as 𝐾𝐾 ( 𝑧𝑧 ) = ½i 𝜋𝜋 [ 𝐽𝐽 (i 𝑧𝑧 ) + i 𝑌𝑌 (i 𝑧𝑧 )] . In cases where 𝑘𝑘 𝑧𝑧 = �𝑘𝑘 − 𝑘𝑘 𝑥𝑥 − 𝑘𝑘 𝑦𝑦 is real-valued, evaluating the Fourier integral yields � exp ( i𝑘𝑘 �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ) �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑥𝑥 d 𝑦𝑦 ∞−∞ = � 𝑒𝑒 i𝑘𝑘 𝑥𝑥 𝑥𝑥 � [ cos ( 𝑘𝑘 �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ) + i sin ( 𝑘𝑘 �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 )] × �cos ( 𝑘𝑘 𝑦𝑦 𝑦𝑦 ) + i sin ( 𝑘𝑘 𝑦𝑦 𝑦𝑦 ) ��𝑥𝑥 +𝑦𝑦 +𝑧𝑧 d 𝑦𝑦 d 𝑥𝑥 ∞𝑦𝑦=−∞∞𝑥𝑥=−∞ = i 𝜋𝜋 � [cos( 𝑘𝑘 𝑥𝑥 𝑥𝑥 ) + i sin( 𝑘𝑘 𝑥𝑥 𝑥𝑥 )] × � 𝐽𝐽 ��𝑘𝑘 − 𝑘𝑘 𝑦𝑦 √𝑥𝑥 + 𝑧𝑧 � + i 𝑌𝑌 ��𝑘𝑘 − 𝑘𝑘 𝑦𝑦 √𝑥𝑥 + 𝑧𝑧 �� d 𝑥𝑥 ∞−∞ = i2 𝜋𝜋 exp � i 𝑧𝑧�𝑘𝑘 − 𝑘𝑘 𝑥𝑥 − 𝑘𝑘 𝑦𝑦 � �𝑘𝑘 − 𝑘𝑘 𝑥𝑥 − 𝑘𝑘 𝑦𝑦 � = i2 𝜋𝜋 exp(i 𝑘𝑘 𝑧𝑧 𝑧𝑧 ) 𝑘𝑘 𝑧𝑧 ⁄ . (E1) If 𝑘𝑘 𝑧𝑧 happens to be imaginary, either because 𝑘𝑘 𝑥𝑥 > 𝑘𝑘 , or 𝑘𝑘 𝑦𝑦 > 𝑘𝑘 or 𝑘𝑘 𝑥𝑥 + 𝑘𝑘 𝑦𝑦 > 𝑘𝑘 , the intermediate steps leading to the final result in Eq.(E1) could be different, but the final result will be the same. The Fourier integral may now be written 𝜋𝜋 exp �−𝑧𝑧�𝑘𝑘 𝑥𝑥 + 𝑘𝑘 𝑦𝑦 − 𝑘𝑘 � �𝑘𝑘 𝑥𝑥 + 𝑘𝑘 𝑦𝑦 − 𝑘𝑘 � . odd function of 𝑥𝑥 odd function of 𝑦𝑦 Appendix F
The Cornu spiral, a graph of the complex Fresnel integral � 𝑒𝑒 i𝜋𝜋𝜁𝜁 d 𝜁𝜁 = 𝐶𝐶 ( 𝑥𝑥 ) + i 𝑆𝑆 ( 𝑥𝑥 ) 𝑥𝑥0 in the complex plane, is shown in Fig.F1. Note that the Fresnel integral as well as its real and imaginary parts, 𝐶𝐶 ( 𝑥𝑥 ) and 𝑆𝑆 ( 𝑥𝑥 ) , are defined differently here than in Eq.(46). Whereas Eq.(46) complies with the standard definition of these functions in accordance with Gradshteyn and Ryzhik’s Table of Integral, Series, and Products , the definitions adopted here are more convenient for describing the spiral. The center of the spiral corresponds to 𝑥𝑥 = 0 , with positive values of 𝑥𝑥 represented on the right arm, and negative values on the left arm of the spiral. Fig. F1 . The Cornu spiral is a complex-plane graph of the Fresnel integral ∫ 𝑒𝑒 i𝜋𝜋𝜁𝜁 d 𝜁𝜁 𝑥𝑥0 = 𝐶𝐶 ( 𝑥𝑥 ) + i 𝑆𝑆 ( 𝑥𝑥 ) . The derivative with respect to 𝑥𝑥 of the complex Fresnel integral is 𝑒𝑒 i𝜋𝜋𝑥𝑥 , indicating that, along the spiral, the points corresponding to 𝑥𝑥 and 𝑥𝑥 + d 𝑥𝑥 are connected by an infinitesimal arrow of length d 𝑥𝑥 and orientation angle 𝜋𝜋𝑥𝑥 ⁄ . (As always, the angle in the complex plane is measured counterclockwise from the positive real axis.) Thus, at 𝑥𝑥 = 0 , the tangent to the spiral is parallel to the real axis; at 𝑥𝑥 = 1 , where the angle of the little arrow is 𝜋𝜋 ⁄ , the tangent is parallel to the imaginary axis; at 𝑥𝑥 = √ , the angle is 𝜋𝜋 and the tangent is anti-parallel to the real axis; at 𝑥𝑥 = 2 , the angle is 𝜋𝜋 and the tangent is, once again, parallel to the real axis. Figure F1 shows a few more points corresponding to other values of 𝑥𝑥 along the length of the spiral. As 𝑥𝑥 → ∞ , the spiral approaches its limit point of ½ + ½i , which is the value of ∫ 𝑒𝑒 i𝜋𝜋𝜁𝜁 d 𝜁𝜁 ∞0 obtained by integrating in the complex 𝜁𝜁 -plane along the line. An asymptotic series for the Fresnel integral can be found by a repeated application of integration-by-parts, as follows: � 𝑒𝑒 i𝜋𝜋𝜁𝜁 d 𝜁𝜁 𝑥𝑥0 = � 𝑒𝑒 i𝜋𝜋𝜁𝜁 d 𝜁𝜁 ∞0 − � 𝑒𝑒 i𝜋𝜋𝜁𝜁 d 𝜁𝜁 ∞𝑥𝑥 = exp ( i𝜋𝜋 4⁄ ) √2 − � (i 𝜋𝜋𝜁𝜁 ) −1 (i 𝜋𝜋𝜁𝜁𝑒𝑒 i𝜋𝜋𝜁𝜁 )d 𝜁𝜁 ∞𝑥𝑥 = exp ( i𝜋𝜋 4⁄ ) √2 + exp ( i𝜋𝜋𝑥𝑥 ) i𝜋𝜋𝑥𝑥 − i 𝜋𝜋 � (i 𝜋𝜋𝜁𝜁 ) −2 𝑒𝑒 i𝜋𝜋𝜁𝜁 d 𝜁𝜁 ∞𝑥𝑥 = exp ( i𝜋𝜋 4⁄ ) √2 + exp ( i𝜋𝜋𝑥𝑥 ) i𝜋𝜋𝑥𝑥 − exp ( i𝜋𝜋𝑥𝑥 ) 𝜋𝜋 𝑥𝑥 + 3 𝜋𝜋 � (i 𝜋𝜋𝜁𝜁 ) −4 𝑒𝑒 i𝜋𝜋𝜁𝜁 d 𝜁𝜁 ∞𝑥𝑥 . (F1) − − − 𝐶𝐶 ( 𝑥𝑥 ) 𝑥𝑥 = 0.5 𝑥𝑥 = − − − − 𝑆𝑆 ( 𝑥𝑥 ) − − − Interpreting the results of diffraction calculations from a half-plane . The Fresnel function 𝐹𝐹 ( 𝜁𝜁 ) = ∫ exp(i 𝑥𝑥 ) d 𝑥𝑥 ∞𝜁𝜁 is related to the Cornu spiral via Eq.(46). The spiral depicted in Fig.F1 represents the function ∫ exp(i 𝜋𝜋𝜁𝜁 ⁄ ) d 𝜁𝜁 𝑥𝑥0 = � 𝜋𝜋⁄ ∫ exp(i 𝑦𝑦 ) d 𝑦𝑦 �𝜋𝜋 2⁄ 𝑥𝑥0 ; consequently, 𝐹𝐹 ( 𝜁𝜁 ) = �𝜋𝜋 ⁄ 𝑒𝑒 i𝜋𝜋 4⁄ − �𝜋𝜋 ⁄ Cornu( � 𝜋𝜋⁄ 𝜁𝜁 ) = �𝜋𝜋 ⁄ � (½ + ½i) − Cornu( � 𝜋𝜋⁄ 𝜁𝜁 ) � . (F2) The complex number representing 𝐹𝐹 ( 𝜁𝜁 ) is thus seen to be �𝜋𝜋 ⁄ times the arrow that goes from the point 𝑥𝑥 = � 𝜋𝜋⁄ 𝜁𝜁 on the Cornu spiral to the spiral’s eye at ½ + ½i . Starting at 𝜁𝜁 = 0 , where 𝐹𝐹 ( 𝜁𝜁 ) = �𝜋𝜋 ⁄ 𝑒𝑒 i𝜋𝜋 4⁄ , when 𝜁𝜁 rises from toward ∞ , the magnitude of 𝐹𝐹 ( 𝜁𝜁 ) shrinks rapidly, while its phase cycles between and 𝜋𝜋 faster and faster. In contrast, when 𝜁𝜁 goes from toward −∞ , the magnitude of 𝐹𝐹 ( 𝜁𝜁 ) quickly grows toward the limit value of √𝜋𝜋𝑒𝑒 i𝜋𝜋 4⁄ , albeit with some oscillations along the way, whereas the phase hovers pretty much around 𝜋𝜋 ⁄ . We can now see how the intensity plot in Fig.3 comes about. In accordance with Eq.(49), the observed intensity is the squared magnitude of 𝐹𝐹��𝜋𝜋 ( 𝜆𝜆 𝑧𝑧 ) ⁄ ( 𝑥𝑥 − 𝑧𝑧 sin 𝜃𝜃 inc ) � . As 𝑥𝑥 rises above 𝑧𝑧 sin 𝜃𝜃 inc into the shadow region, the magnitude of 𝐹𝐹 ( ∙ ) quickly drops from ½ √𝜋𝜋 to zero. However, when 𝑥𝑥 goes below 𝑧𝑧 sin 𝜃𝜃 inc , the magnitude of 𝐹𝐹 ( ∙ ) rises rapidly — and with rapid oscillations — toward the limiting value of √𝜋𝜋 . One can similarly interpret Sommerfeld’s famous result in Eq.(81) pertaining to diffraction from a thin, perfectly conducting half-plane. The plane-wave 𝑒𝑒 i𝑘𝑘 𝑘𝑘 cos ( 𝜃𝜃−𝜃𝜃 ) appearing in Eq.(81) is the incident plane-wave. In the system depicted in Fig.5, consider a circle in the 𝑥𝑥𝑧𝑧 -plane centered at the origin of coordinates, whose radius 𝑟𝑟 is large enough to make � 𝑘𝑘 𝑟𝑟 a fairly large number. If the observation point 𝒓𝒓 = ( 𝑟𝑟 , 𝜃𝜃 ) is located in the shadow region, where ≤ 𝜃𝜃 < 𝜃𝜃 , the argument of 𝐹𝐹�−� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 − 𝜃𝜃 ) 2 ⁄ ] � will be large and positive, making the amplitude of the incident plane-wave in the shadow region exceedingly small. Near the edge of the shadow, where 𝜃𝜃 ≅ 𝜃𝜃 , the magnitude of 𝐹𝐹 ( ∙ ) is ~½ √𝜋𝜋 , and the shadow begins to recede. In the remaining part of the circle of radius 𝑟𝑟 , where 𝜃𝜃 < 𝜃𝜃 ≤ 𝜋𝜋 , the argument of 𝐹𝐹 ( ∙ ) will be large and negative, indicating the presence of the incident plane-wave at essentially its full strength. The plane-wave 𝑒𝑒 i𝑘𝑘 𝑘𝑘 cos ( 𝜃𝜃+𝜃𝜃 ) , which is also present in Eq.(81), corresponds to the reflected wave at the front facet of the perfectly conducting mirror — because the angle of the reflected 𝑘𝑘 -vector with the 𝑥𝑥 -axis is −𝜃𝜃 . Now, on the aforementioned circle of radius 𝑟𝑟 , the argument of the coefficient 𝐹𝐹�� 𝑘𝑘 𝑟𝑟 sin[( 𝜃𝜃 + 𝜃𝜃 ) 2 ⁄ ] � of the reflected wave is large and positive in the region where ≤ 𝜃𝜃 < 2 𝜋𝜋 − 𝜃𝜃 . Therefore, in this region, the contribution of the reflected wave to the overall observed 𝐸𝐸 -field is negligible. The situation changes drastically as 𝜃𝜃 approaches 𝜋𝜋 − 𝜃𝜃 , at which point the magnitude of 𝐹𝐹 ( ∙ ) climbs to ½ √𝜋𝜋 , and then continues to grow — with characteristic oscillations — as 𝜃𝜃 goes from 𝜋𝜋 − 𝜃𝜃 to 𝜋𝜋 . Note that at 𝜃𝜃 = 0 and 𝜋𝜋 , irrespective of the distance 𝑟𝑟 from the edge, the two plane-waves in Eq.(81), being exactly equal and opposite, cancel out, as required by the boundary condition at the front and back facets of the mirror. The incident and reflected waves interfere in the region 𝜋𝜋 − 𝜃𝜃 ≲ 𝜃𝜃 ≤ 𝜋𝜋 , where both waves have a strong presence. The oscillations occur near the edge of the shadow, where 𝜃𝜃 ≅ 𝜃𝜃 , and also near the edge of the reflected beam, 𝜃𝜃 ≅ 𝜋𝜋 − 𝜃𝜃 , where the argument of the coefficients 𝐹𝐹 ( ∙ ) transitions from positive to negative values. 5 Appendix G
We evaluate the integral that appears in Eq.(76). In what follows, 𝜆𝜆 is real and positive, whereas 𝜂𝜂 , while real, may be positive or negative. We use the fact that � 𝑒𝑒 −𝜆𝜆𝜁𝜁 d 𝜁𝜁 ∞0 = ½ �𝜋𝜋 𝜆𝜆⁄ . � exp ( −𝜆𝜆𝜁𝜁 ) 𝜁𝜁 − i𝜂𝜂 d 𝜁𝜁 ∞0 = 𝑒𝑒 −i𝜆𝜆𝜂𝜂 � exp [ −𝜆𝜆 ( 𝜁𝜁 − i𝜂𝜂 )] 𝜁𝜁 − i𝜂𝜂 d 𝜁𝜁 ∞0 = −𝑒𝑒 −i𝜆𝜆𝜂𝜂 � � dd𝜆𝜆 � exp [ −𝜆𝜆 ( 𝜁𝜁 − i𝜂𝜂 )] 𝜁𝜁 − i𝜂𝜂 d 𝜁𝜁 ∞𝜁𝜁=0 � d 𝜆𝜆 ∞𝜆𝜆 = 𝑒𝑒 −i𝜆𝜆𝜂𝜂 � �� 𝑒𝑒 −𝜆𝜆 ( 𝜁𝜁 −i𝜂𝜂 ) d 𝜁𝜁 ∞𝜁𝜁=0 � d 𝜆𝜆 ∞𝜆𝜆 = 𝑒𝑒 −i𝜆𝜆𝜂𝜂 � 𝑒𝑒 i𝜆𝜆𝜂𝜂 �� 𝑒𝑒 −𝜆𝜆𝜁𝜁 d 𝜁𝜁 ∞0 � d 𝜆𝜆 ∞𝜆𝜆 = ½ √𝜋𝜋𝑒𝑒 −i𝜆𝜆𝜂𝜂 � 𝜆𝜆 − ½ 𝑒𝑒 i𝜆𝜆𝜂𝜂 d 𝜆𝜆 ∞𝜆𝜆 = √𝜋𝜋 | 𝜂𝜂 | −1 𝑒𝑒 −i𝜆𝜆𝜂𝜂 � 𝑒𝑒 i𝑥𝑥 d 𝑥𝑥 ∞ | 𝜂𝜂 | √𝜆𝜆 = √𝜋𝜋 | 𝜂𝜂 | −1 𝑒𝑒 −i𝜆𝜆𝜂𝜂 𝐹𝐹 (| 𝜂𝜂 | √𝜆𝜆 ) . (G1) In this equation, 𝐹𝐹 ( 𝛼𝛼 ) = ∫ exp(i 𝑥𝑥 ) d 𝑥𝑥 ∞𝛼𝛼 is the complex Fresnel integral as defined in Eq.(46). Appendix H
In the case of −𝜋𝜋 ≤ 𝜃𝜃 ≤ , the scattered 𝐸𝐸 -field can be computed directly using the integration path shown in Fig. H1. The blue contour corresponds to the path shown in Fig.6(b), where cos 𝜑𝜑 = 𝜎𝜎 𝑥𝑥 goes from −∞ to ∞ . Note that 𝜎𝜎 𝑧𝑧 = � − 𝜎𝜎 𝑥𝑥 = sin( 𝜑𝜑 ′ + i 𝜑𝜑 ″ ) is negative imaginary on the vertical legs of the contour, and negative real on its horizontal leg. This is a necessary condition because 𝑧𝑧 is now negative whereas 𝜎𝜎 𝑧𝑧 𝑧𝑧 must remain positive. The pole at 𝜎𝜎 𝑥𝑥 = cos 𝜃𝜃 now appears at 𝜑𝜑 = −𝜃𝜃 , and the small semi-circle around the pole in the 𝜎𝜎 𝑥𝑥 -plane now becomes a small dip in the vicinity of 𝜑𝜑 = −𝜃𝜃 . The remaining factor in Eq.(68) is � 𝜎𝜎 𝑥𝑥 , which retains its identity when replaced by cos( 𝜑𝜑 ⁄ ) on the blue trajectory of Fig. H1. The saddle point is at 𝜑𝜑 = 𝜃𝜃 , and the pole at 𝜑𝜑 = −𝜃𝜃 needs to be accounted for only when 𝜃𝜃 > −𝜃𝜃 . Fig. H1 . Shown in blue is the integration path in the complex 𝜑𝜑 -plane for the range −𝜋𝜋 ≤ 𝜃𝜃 ≤ . The corresponding steepest-descent contour 𝑆𝑆 ( 𝜃𝜃 ) is shown in red. The small dip around 𝜑𝜑 = −𝜃𝜃 corresponds to the small semi-circle around 𝜎𝜎 𝑥𝑥 = cos 𝜃𝜃 in the contour depicted in Fig.6(b). 𝑥𝑥 = | 𝜂𝜂 | √𝜆𝜆 𝜑𝜑 ′ −𝜋𝜋 𝜑𝜑 ″ −𝜃𝜃 𝜃𝜃 𝜃𝜃 + ½ 𝜋𝜋 𝜃𝜃 − ½ 𝜋𝜋 𝑆𝑆 ( 𝜃𝜃 ) Appendix I
We evaluate the integral in Eq.(121) which, upon straightforward algebraic manipulation, leads to 𝑓𝑓 ( 𝒒𝒒 ) = 𝓂𝓂 n 𝜇𝜇 ℏ �𝝁𝝁 n ∙ 𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) � (8 𝜋𝜋 ⁄ ) − ∫ 𝑟𝑟 −3 exp(i 𝒒𝒒 ∙ 𝒓𝒓 ℏ⁄ ) d 𝒓𝒓 ∞−∞ � +3 � ( 𝝁𝝁 n ∙ 𝒓𝒓� ) �𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) ∙ 𝒓𝒓��𝑟𝑟 −3 exp(i 𝒒𝒒 ∙ 𝒓𝒓 ℏ⁄ ) d 𝒓𝒓 ∞−∞ � . (I1) With reference to Fig.I1, the first integral appearing in Eq.(I1) is readily evaluated as follows: ∫ 𝑟𝑟 −3 exp(i 𝒒𝒒 ∙ 𝒓𝒓 ℏ⁄ ) d 𝑟𝑟 ∞−∞ = � ∫ 𝑟𝑟 −3 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ ) (2 𝜋𝜋𝑟𝑟 sin 𝜃𝜃 )d 𝜃𝜃 d 𝑟𝑟 𝜋𝜋𝜃𝜃=0∞𝑟𝑟=0 = 2 𝜋𝜋 � exp ( i𝑞𝑞𝑘𝑘 ℏ⁄ ) − exp ( −i𝑞𝑞𝑘𝑘 ℏ⁄ ) i𝑞𝑞𝑘𝑘 ℏ⁄ d 𝑟𝑟 ∞𝑟𝑟=0 = 4 𝜋𝜋 � sin ( 𝑞𝑞𝑘𝑘 ℏ⁄ ) 𝑞𝑞𝑘𝑘 ℏ⁄ d 𝑟𝑟 ∞0 = 4 𝜋𝜋 ∫ (sin 𝑥𝑥 𝑥𝑥 ⁄ )d 𝑥𝑥 ∞0 = 4 𝜋𝜋�− (sin 𝑥𝑥 𝑥𝑥⁄ )| + ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 � = 4 𝜋𝜋� ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 � . (I2) Fig.I1 . In the 3-dimensional Cartesian space of 𝒓𝒓 = 𝑥𝑥𝒙𝒙� + 𝑦𝑦𝒚𝒚� + 𝑧𝑧𝒛𝒛� , the integrands in Eq.(I1) have certain symmetries around the frequency vector 𝒒𝒒 . The angle between 𝒒𝒒 and 𝒓𝒓 is denoted by 𝜃𝜃 , while the azimuthal angle around the 𝑞𝑞 -axis is 𝜑𝜑 (not shown). The vectors 𝝁𝝁 n and 𝑴𝑴� have projections 𝝁𝝁 ⊥ and 𝑴𝑴� ⊥ in the plane perpendicular to the unit-vector 𝒒𝒒� (associated with the spatial frequency 𝒒𝒒 ). The projection of 𝒓𝒓 onto the plane perpendicular to 𝒒𝒒� is along the unit-vector 𝝆𝝆� . As for the remaining integral, we write 𝝁𝝁 n , 𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) , 𝒓𝒓� as the sum of their projections along the unit-vector 𝒒𝒒� and onto the plane orthogonal to 𝒒𝒒� . Thus, 𝝁𝝁 n = ( 𝝁𝝁 n ∙ 𝒒𝒒� ) 𝒒𝒒� + 𝝁𝝁 ⊥ , and 𝑴𝑴� =( 𝑴𝑴� ∙ 𝒒𝒒� ) 𝒒𝒒� + 𝑴𝑴� ⊥ , while 𝒓𝒓� = (cos 𝜃𝜃 ) 𝒒𝒒� + (sin 𝜃𝜃 ) 𝝆𝝆� , where 𝝆𝝆� , the unit-vector along the projection of 𝒓𝒓� in the plane orthogonal to 𝒒𝒒� , makes the angles 𝜑𝜑 𝜇𝜇 and 𝜑𝜑 𝑀𝑀 with 𝝁𝝁 ⊥ and 𝑴𝑴� ⊥ . We will have ( 𝝁𝝁 n ∙ 𝒓𝒓� )( 𝑴𝑴� ∙ 𝒓𝒓� ) = ( 𝝁𝝁 n ∙ 𝒒𝒒� cos 𝜃𝜃 + 𝜇𝜇 ⊥ sin 𝜃𝜃 cos 𝜑𝜑 𝜇𝜇 )( 𝑴𝑴� ∙ 𝒒𝒒� cos 𝜃𝜃 + 𝑀𝑀� ⊥ sin 𝜃𝜃 cos 𝜑𝜑 𝑀𝑀 ) = ( 𝝁𝝁 n ∙ 𝒒𝒒� )( 𝑴𝑴� ∙ 𝒒𝒒� ) cos 𝜃𝜃 + ( 𝝁𝝁 n ∙ 𝒒𝒒� ) 𝑀𝑀� ⊥ sin 𝜃𝜃 cos 𝜃𝜃 cos 𝜑𝜑 𝑀𝑀 +( 𝑴𝑴� ∙ 𝒒𝒒� ) 𝜇𝜇 ⊥ sin 𝜃𝜃 cos 𝜃𝜃 cos 𝜑𝜑 𝜇𝜇 + ½ 𝜇𝜇 ⊥ 𝑀𝑀� ⊥ sin 𝜃𝜃 cos( 𝜑𝜑 𝜇𝜇 + 𝜑𝜑 𝑀𝑀 ) +½ 𝜇𝜇 ⊥ 𝑀𝑀� ⊥ sin 𝜃𝜃 cos( 𝜑𝜑 𝜇𝜇 − 𝜑𝜑 𝑀𝑀 ) . (I3) 𝑥𝑥 𝑧𝑧 𝑦𝑦 𝒓𝒓 𝒒𝒒 𝜃𝜃 𝒒𝒒� 𝝁𝝁 n 𝝁𝝁 ⊥ 𝑴𝑴�
𝑴𝑴� ⊥ 𝝆𝝆 � 𝜑𝜑 𝑀𝑀 𝜑𝜑 𝜇𝜇 The divergent term will eventually cancel out. 𝜑𝜑 causes the 2 nd , 3 rd and 4 th terms (colored blue) on the right-hand side of Eq.(I3) to vanish. The remaining terms now yield � ( 𝝁𝝁 n ∙ 𝒓𝒓� ) �𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) ∙ 𝒓𝒓��𝑟𝑟 −3 exp(i 𝒒𝒒 ∙ 𝒓𝒓 ℏ⁄ ) d 𝒓𝒓 ∞−∞ = ( 𝝁𝝁 n ∙ 𝒒𝒒� )( 𝑴𝑴� ∙ 𝒒𝒒� ) � 𝑟𝑟 −3 ∫ cos 𝜃𝜃 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ ) (2 𝜋𝜋𝑟𝑟 sin 𝜃𝜃 )d 𝜃𝜃 d 𝑟𝑟 𝜋𝜋𝜃𝜃=0∞𝑟𝑟=0 +½( 𝝁𝝁 ⊥ ∙ 𝑴𝑴� ⊥ ) � 𝑟𝑟 −3 ∫ sin 𝜃𝜃 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ ) (2 𝜋𝜋𝑟𝑟 sin 𝜃𝜃 )d 𝜃𝜃 d 𝑟𝑟 𝜋𝜋𝜃𝜃=0∞𝑟𝑟=0 = 2 𝜋𝜋 [( 𝝁𝝁 n ∙ 𝒒𝒒� )( 𝑴𝑴� ∙ 𝒒𝒒� ) − ½( 𝝁𝝁 ⊥ ∙ 𝑴𝑴� ⊥ )] � 𝑟𝑟 −1 ∫ sin 𝜃𝜃 cos 𝜃𝜃 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ ) d 𝜃𝜃 d 𝑟𝑟 𝜋𝜋𝜃𝜃=0∞𝑟𝑟=0 +2 𝜋𝜋 ( 𝝁𝝁 ⊥ ∙ 𝑴𝑴� ⊥ ) � ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 � . (I4) Next, we evaluate the 2D integral on the right-hand side of Eq.(I4), as follows: � 𝑟𝑟 −1 ∫ sin 𝜃𝜃 cos 𝜃𝜃 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ ) d 𝜃𝜃 d 𝑟𝑟 𝜋𝜋𝜃𝜃=0∞𝑟𝑟=0 = � 𝑟𝑟 −1 � iℏ𝑞𝑞𝑘𝑘 � [cos 𝜃𝜃 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ )| 𝜃𝜃=0𝜋𝜋∞𝑟𝑟=0 +2 ∫ sin 𝜃𝜃 cos 𝜃𝜃 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ ) d 𝜃𝜃 𝜋𝜋𝜃𝜃=0 � d 𝑟𝑟 = � �
2ℏ sin ( 𝑞𝑞𝑘𝑘 ℏ⁄ ) 𝑞𝑞𝑘𝑘 + � � � iℏ𝑞𝑞𝑘𝑘 � � cos 𝜃𝜃 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ )| 𝜃𝜃=0𝜋𝜋 + ∫ sin 𝜃𝜃 exp(i 𝑞𝑞𝑟𝑟 cos 𝜃𝜃 ℏ⁄ ) d 𝜃𝜃 𝜋𝜋0 �� d 𝑟𝑟 ∞𝑟𝑟=0 = � �
2ℏ sin ( 𝑞𝑞𝑘𝑘 ℏ⁄ ) 𝑞𝑞𝑘𝑘 + 𝑞𝑞 𝑘𝑘 � 𝑞𝑞𝑟𝑟 ℏ⁄ ) −
2ℏ sin ( 𝑞𝑞𝑘𝑘 ℏ⁄ ) 𝑞𝑞𝑘𝑘 �� d 𝑟𝑟 ∞0 = � � + − � d 𝑥𝑥 ∞0 = 2 ∫ (sin 𝑥𝑥 𝑥𝑥 ⁄ )d 𝑥𝑥 ∞0 + 4 ∫ (cos 𝑥𝑥 𝑥𝑥 ⁄ )d 𝑥𝑥 ∞0 + (4 sin 𝑥𝑥 𝑥𝑥 ⁄ )| − (4 3 ⁄ ) ∫ (cos 𝑥𝑥 𝑥𝑥 ⁄ )d 𝑥𝑥 ∞0 = (4 sin 𝑥𝑥 𝑥𝑥 ⁄ )| + 2 ∫ (sin 𝑥𝑥 𝑥𝑥 ⁄ )d 𝑥𝑥 ∞0 − (4 cos 𝑥𝑥 𝑥𝑥 ⁄ )| − ∫ (4 sin 𝑥𝑥 𝑥𝑥 ⁄ )d 𝑥𝑥 ∞0 = (4 sin 𝑥𝑥 𝑥𝑥 ⁄ )| − (4 cos 𝑥𝑥 𝑥𝑥 ⁄ )| − ⅔ (sin 𝑥𝑥 𝑥𝑥⁄ )| + ⅔ ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 = − � − ! + ⋯ � 𝑥𝑥→0 + � − ! + ⋯ � 𝑥𝑥→0 + + ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 = + ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 . (I5) Recognizing that ( 𝝁𝝁 n ∙ 𝒒𝒒� )( 𝑴𝑴� ∙ 𝒒𝒒� ) + ( 𝝁𝝁 ⊥ ∙ 𝑴𝑴� ⊥ ) = 𝝁𝝁 n ∙ 𝑴𝑴� , Eq.(I4) becomes � ( 𝝁𝝁 n ∙ 𝒓𝒓� ) �𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) ∙ 𝒓𝒓��𝑟𝑟 −3 exp(i ℏ𝒒𝒒 ∙ 𝒓𝒓 ) d 𝒓𝒓 ∞−∞ = � ( 𝝁𝝁 n ∙ 𝒒𝒒� )( 𝑴𝑴� ∙ 𝒒𝒒� ) + 4( 𝝁𝝁 ⊥ ∙ 𝑴𝑴� ⊥ ) � + ( 𝝁𝝁 n ∙ 𝑴𝑴� ) ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 . (I6) Finally, upon substituting from Eqs.(I2) and (I6) into Eq.(I1), we arrive at 𝑓𝑓 ( 𝒒𝒒 ) = 𝓂𝓂 n 𝜇𝜇 ℏ �𝝁𝝁 n ∙ 𝑴𝑴� � (8 𝜋𝜋 ⁄ ) − 𝜋𝜋 − 𝜋𝜋 ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 � +(4 𝜋𝜋 ⁄ ) � ( 𝝁𝝁 n ∙ 𝒒𝒒� )( 𝑴𝑴� ∙ 𝒒𝒒� ) + 4( 𝝁𝝁 ⊥ ∙ 𝑴𝑴� ⊥ ) � + 4 𝜋𝜋𝝁𝝁 n ∙ 𝑴𝑴� ∫ (cos 𝑥𝑥 𝑥𝑥⁄ )d 𝑥𝑥 ∞0 � = � 𝓂𝓂 n ℏ � 𝝁𝝁 n ∙ �𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) − �𝑴𝑴� ( 𝒒𝒒 ℏ⁄ ) ∙ 𝒒𝒒��𝒒𝒒�� . (I7) This is the same result as given in Ref. [29], Eq.(23), in the case of 𝜆𝜆 = 1 . The coefficient 𝜋𝜋𝜇𝜇 appears here because we have worked in the 𝑆𝑆𝑆𝑆 system of units with 𝑩𝑩 = 𝜇𝜇 𝑯𝑯 + 𝑴𝑴 . The divergent term will eventually cancel out. Appendix J
To establish the equivalence of the Rayleigh-Sommerfeld diffraction formulation with the plane-wave expansion method based on Fourier transformation of the initial distribution in the 𝑥𝑥𝑦𝑦 -plane at 𝑧𝑧 = 0 , we begin by differentiating both sides of Eq.(37) with respect to 𝑧𝑧 , as follows: � 𝜕𝜕 𝑧𝑧 � exp ( i𝑘𝑘 �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ) �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 � 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑥𝑥 d 𝑦𝑦 ∞−∞ = − 𝜋𝜋 exp(i 𝑘𝑘 𝑧𝑧 𝑧𝑧 ) . (J1) Here, 𝑘𝑘 𝑧𝑧 = �𝑘𝑘 − 𝑘𝑘 𝑥𝑥 − 𝑘𝑘 𝑦𝑦 could be real or imaginary, depending on whether 𝑘𝑘 𝑥𝑥 + 𝑘𝑘 𝑦𝑦 is less than or greater than 𝑘𝑘 . An inverse Fourier transformation now yields (2 𝜋𝜋 ) −2 � 𝑒𝑒 i𝑘𝑘 𝑧𝑧 𝑧𝑧 𝑒𝑒 −i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ = − 𝜕𝜕 𝑧𝑧 � exp ( i𝑘𝑘 �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ) �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 � . (J2) The properly defined inverse transform of exp(i 𝑘𝑘 𝑧𝑧 𝑧𝑧 ) is obtained by reversing the signs of 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 in Eq.(J2), which is tantamount to flipping the signs of 𝑥𝑥 and 𝑦𝑦 . Consequently, ℱ −1 �𝑒𝑒 i𝑘𝑘 𝑧𝑧 𝑧𝑧 � = (2 𝜋𝜋 ) −2 � 𝑒𝑒 i𝑘𝑘 𝑧𝑧 𝑧𝑧 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ = − 𝜕𝜕 𝑧𝑧 � exp ( i𝑘𝑘 �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ) �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 � = 𝑧𝑧2𝜋𝜋 ( 𝑥𝑥 +𝑦𝑦 +𝑧𝑧 ) � +𝑦𝑦 +𝑧𝑧 − i 𝑘𝑘 � 𝑒𝑒 i𝑘𝑘 �𝑥𝑥 +𝑦𝑦 +𝑧𝑧 . (J3) The above formula is associated with the name of Hermann Weyl. Suppose now that the scalar field 𝜓𝜓 in the 𝑥𝑥𝑦𝑦 -plane at 𝑧𝑧 = 0 is represented by its Fourier spectrum 𝜓𝜓� , namely, 𝜓𝜓 ( 𝑥𝑥 , 𝑦𝑦 , 𝑧𝑧 = 0) = (2 𝜋𝜋 ) −2 ∬ 𝜓𝜓� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦 ∞−∞ . (J4) Propagating this field by a distance 𝑧𝑧 along the 𝑧𝑧 -axis requires the multiplication of 𝜓𝜓� by the propagation factor exp(i 𝑘𝑘 𝑧𝑧 𝑧𝑧 ) . Recalling that the Fourier transform of the convolution of two functions is the product of the individual transforms of those functions, † we will have 𝜓𝜓 ( 𝑥𝑥 , 𝑦𝑦 , 𝑧𝑧 ) = ℱ −1 �𝜓𝜓� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) 𝑒𝑒 i𝑘𝑘 𝑧𝑧 𝑧𝑧 � = 𝜓𝜓 ( 𝑥𝑥 , 𝑦𝑦 , 𝑧𝑧 = 0) ∗ ℱ −1 �𝑒𝑒 i𝑘𝑘 𝑧𝑧 𝑧𝑧 � = − � 𝜓𝜓 ( 𝑥𝑥 , 𝑦𝑦 , 𝑧𝑧 = 0) 𝜕𝜕 𝑧𝑧 � exp�i𝑘𝑘 [( 𝑥𝑥 −𝑥𝑥 ) + ( 𝑦𝑦 −𝑦𝑦 ) +𝑧𝑧 ] ½ � [( 𝑥𝑥 −𝑥𝑥 ) + ( 𝑦𝑦 −𝑦𝑦 ) +𝑧𝑧 ] ½ � d 𝑥𝑥 d 𝑦𝑦 ∞−∞ . (J5) This is the same as the Rayleigh-Sommerfeld formula given in Eq.(25), with 𝑆𝑆 being the 𝑥𝑥𝑦𝑦 -plane at 𝑧𝑧 = 0 , and ( 𝒓𝒓 − 𝒓𝒓 ) ∙ 𝒏𝒏� being equal to −𝑧𝑧 . In similar fashion, one can relate Eqs.(32) and (33) to their Fourier method counterparts. For instance, recognizing the integral in Eq.(32) as a convolution integral, we use Eq.(37) to write † Convolution theorem: ℱ −1 �𝐹𝐹� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) 𝐺𝐺� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) � = (2 𝜋𝜋 ) −2 � 𝐹𝐹� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) 𝐺𝐺� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ = (2 𝜋𝜋 ) −2 � �∬ 𝐹𝐹 ( 𝑥𝑥 ′ , 𝑦𝑦 ′ ) 𝑒𝑒 −i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥 ′ +𝑘𝑘 𝑦𝑦 𝑦𝑦 ′ ) d 𝑥𝑥 ′ d 𝑦𝑦 ′∞−∞ �𝐺𝐺� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥+𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ = � 𝐹𝐹 ( 𝑥𝑥 ′ , 𝑦𝑦 ′ ) � (2 𝜋𝜋 ) −2 ∬ 𝐺𝐺� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) 𝑒𝑒 i [ 𝑘𝑘 𝑥𝑥 ( 𝑥𝑥−𝑥𝑥 ′ ) +𝑘𝑘 𝑦𝑦 ( 𝑦𝑦−𝑦𝑦 ′ )] d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ � d 𝑥𝑥 ′ d 𝑦𝑦 ′∞−∞ = ∬ 𝐹𝐹 ( 𝑥𝑥 ′ , 𝑦𝑦 ′ ) 𝐺𝐺 ( 𝑥𝑥 − 𝑥𝑥 ′ , 𝑦𝑦 − 𝑦𝑦 ′ )d 𝑥𝑥 ′ d 𝑦𝑦 ′∞−∞ = 𝐹𝐹 ( 𝑥𝑥 , 𝑦𝑦 ) ∗ 𝐺𝐺 ( 𝑥𝑥 , 𝑦𝑦 ) . 𝑬𝑬 ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −1 𝜵𝜵 × � [ 𝒛𝒛� × 𝑬𝑬 ( 𝑥𝑥 , 𝑦𝑦 , 𝑧𝑧 = 0)] exp { i𝑘𝑘 [( 𝑥𝑥 −𝑥𝑥 ) + ( 𝑦𝑦 −𝑦𝑦 ) +𝑧𝑧 ] ½ }[( 𝑥𝑥 −𝑥𝑥 ) + ( 𝑦𝑦 −𝑦𝑦 ) +𝑧𝑧 ] ½ d 𝑥𝑥 d 𝑦𝑦 ∞−∞ = (2 𝜋𝜋 ) −1 𝜵𝜵 × (2 𝜋𝜋 ) −2 � [ 𝒛𝒛� × 𝑬𝑬� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 )](i2 𝜋𝜋𝑒𝑒 i𝑘𝑘 𝑧𝑧 𝑧𝑧 𝑘𝑘 𝑧𝑧 ⁄ ) 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥 +𝑘𝑘 𝑦𝑦 𝑦𝑦 ) d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ = (2 𝜋𝜋 ) −2 � i ( 𝑘𝑘 𝑥𝑥 𝒙𝒙� + 𝑘𝑘 𝑦𝑦 𝒚𝒚� + 𝑘𝑘 𝑧𝑧 𝒛𝒛� ) × ( 𝒛𝒛� × 𝑬𝑬� ) 𝑘𝑘 𝑧𝑧−1 𝑒𝑒 i ( 𝑘𝑘 𝑥𝑥 𝑥𝑥 +𝑘𝑘 𝑦𝑦 𝑦𝑦 +𝑘𝑘 𝑧𝑧 𝑧𝑧 ) d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ = (2 𝜋𝜋 ) −2 � [( 𝒌𝒌 ∙ 𝒛𝒛� ) 𝑬𝑬� − ( 𝒌𝒌 ∙ 𝑬𝑬� ) 𝒛𝒛� ] 𝑘𝑘 𝑧𝑧−1 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ = (2 𝜋𝜋 ) −2 � 𝑬𝑬� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) 𝑒𝑒 i𝒌𝒌 ∙ 𝒓𝒓 d 𝑘𝑘 𝑥𝑥 d 𝑘𝑘 𝑦𝑦∞−∞ . (J6) This, of course, is the expression of the Fourier method of diffraction calculation, in which 𝐸𝐸 𝑥𝑥 ( 𝑥𝑥 , 𝑦𝑦 , 𝑧𝑧 = 0) and 𝐸𝐸 𝑦𝑦 ( 𝑥𝑥 , 𝑦𝑦 , 𝑧𝑧 = 0) are Fourier transformed within the 𝑥𝑥𝑦𝑦 -plane of the initial distribution to yield 𝐸𝐸� 𝑥𝑥 ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) and 𝐸𝐸� 𝑦𝑦 ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) . Using 𝒌𝒌 = 𝑘𝑘 𝑥𝑥 𝒙𝒙� + 𝑘𝑘 𝑦𝑦 𝒚𝒚� + �𝑘𝑘 − 𝑘𝑘 𝑥𝑥 − 𝑘𝑘 𝑦𝑦 𝒛𝒛� and invoking Maxwell’s first equation 𝑘𝑘 𝑥𝑥 𝐸𝐸� 𝑥𝑥 + 𝑘𝑘 𝑦𝑦 𝐸𝐸� 𝑦𝑦 + 𝑘𝑘 𝑧𝑧 𝐸𝐸� 𝑧𝑧 = 0 to determine 𝐸𝐸� 𝑧𝑧 , one arrives at the full expression for 𝑬𝑬� ( 𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 ) , from which 𝑬𝑬 ( 𝒓𝒓 ) is computed via Eq.(J6). Appendix K
The rigorous version of Babinet’s principle of complementary screens pertains to a system such as that of Fig.K1(a) and its complement depicted in Fig.K1(b). The screen in Fig.K1(a) is a perfectly conducting thin sheet located in the 𝑥𝑥𝑦𝑦 -plane at 𝑧𝑧 = 0 . The incident beam passes through the aperture and is observed at 𝒓𝒓 . The observed 𝐸𝐸 -field is computed from Eq.(32), whose domain of integration is now confined to the area of the aperture — since the tangential 𝐸𝐸 -field on the metallic screen vanishes. We will have 𝑬𝑬 ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −1 𝜵𝜵 × ∫ [ 𝒛𝒛� × 𝑬𝑬 ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 aperture . (K1) In the complementary setup of Fig.K1(b), the screen is removed and a thin, perfectly conducting obstacle of the same shape as the aperture of Fig.K1(a) is placed at the aperture’s location. Also, the incident 𝑬𝑬 and 𝑩𝑩 fields are changed to 𝑬𝑬 inc ( c ) = 𝑐𝑐𝑩𝑩 inc and 𝑩𝑩 inc ( c ) = −𝑬𝑬 inc 𝑐𝑐⁄ . We construct the new fields 𝓔𝓔 ( 𝒓𝒓 ) = 𝑬𝑬 ( 𝒓𝒓 ) − 𝑐𝑐𝑩𝑩 ( c ) ( 𝒓𝒓 ) and 𝓑𝓑 ( 𝒓𝒓 ) = 𝑩𝑩 ( 𝒓𝒓 ) + 𝑐𝑐 −1 𝑬𝑬 ( c ) ( 𝒓𝒓 ) in the half-space 𝑧𝑧 ≥ . It is not difficult to verify that 𝓔𝓔 and 𝓑𝓑 satisfy all four of Maxwell’s equations in this empty half-space. Now, in the system of Fig.K1(a), the tangential 𝑬𝑬 is zero on the metallic surface, while in the aperture, the tangential 𝑩𝑩 equals 𝑩𝑩 inc ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) . ‡ Similarly, in the system of Fig.K1(b), the tangential 𝑬𝑬 ( c ) on the metallic surface vanishes, while in the open areas, the tangential 𝑩𝑩 ( c ) equals 𝑩𝑩 inc ∥ ( c ) ( 𝑥𝑥 , 𝑦𝑦 , 0) , which is the same as −𝑬𝑬 inc ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) 𝑐𝑐⁄ . It is thus seen that, on the metallic surface of Fig.K1(a), 𝓔𝓔 ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) = 𝑬𝑬 inc ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) , while in the corresponding ‡ The reason that the tangential 𝑩𝑩 in the aperture equals 𝑩𝑩 inc ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) is that, within the aperture area, the tangential component 𝑩𝑩 𝑠𝑠∥ of the scattered 𝐵𝐵 -field vanishes. To see this, observe that the scattered 𝑬𝑬 and 𝑩𝑩 fields are produced by the induced surface charge- and current-densities in the perfectly conducting regions of the screen 𝑆𝑆 . Since the surface current-density does not have a 𝑧𝑧 -component, the only components of the scattered vector potential 𝑨𝑨 𝑠𝑠 ( 𝒓𝒓 ) will be 𝐴𝐴 𝑠𝑠𝑥𝑥 and 𝐴𝐴 𝑠𝑠𝑦𝑦 . The symmetry of the system under consideration then ensures that, within the aperture area, 𝜕𝜕 𝑧𝑧 𝐴𝐴 𝑠𝑠𝑥𝑥 = 0 and 𝜕𝜕 𝑧𝑧 𝐴𝐴 𝑠𝑠𝑦𝑦 = 0 , which results in 𝐵𝐵 𝑠𝑠𝑥𝑥 = 𝐵𝐵 𝑠𝑠𝑦𝑦 = 0 . (A similar argument reveals that 𝐸𝐸 𝑠𝑠𝑧𝑧 within the aperture vanishes as well, since 𝜕𝜕 𝑡𝑡 𝐴𝐴 𝑠𝑠𝑧𝑧 = 0 and the symmetry of the scattered scalar potential 𝜓𝜓 𝑠𝑠 ( 𝒓𝒓 ) ensures that 𝜕𝜕 𝑧𝑧 𝜓𝜓 𝑠𝑠 = 0 .) 0 𝓑𝓑 ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) = 𝑩𝑩 inc ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) . With these boundary conditions, the solution of Maxwell’s equations for 𝓔𝓔 and 𝓑𝓑 in the half-space 𝑧𝑧 ≥ will be 𝓔𝓔 ( 𝒓𝒓 ) = 𝑬𝑬 inc ( 𝒓𝒓 ) and 𝓑𝓑 ( 𝒓𝒓 ) = 𝑩𝑩 inc ( 𝒓𝒓 ) . § We thus have 𝑬𝑬 ( 𝒓𝒓 ) − 𝑐𝑐𝑩𝑩 ( c ) ( 𝒓𝒓 ) = 𝑬𝑬 inc ( 𝒓𝒓 ) . (K2) 𝑩𝑩 ( 𝒓𝒓 ) + 𝑐𝑐 −1 𝑬𝑬 ( c ) ( 𝒓𝒓 ) = 𝑩𝑩 inc ( 𝒓𝒓 ) . (K3) Fig. K1 . (a) Sitting in the 𝑥𝑥𝑦𝑦 -plane at 𝑧𝑧 = 0 is a thin, perfectly conducting screen containing one or more apertures. The incident beam, having electric field 𝑬𝑬 inc ( 𝒓𝒓 ) and magnetic field 𝑩𝑩 inc ( 𝒓𝒓 ) , arrives from the left-hand side. The observation point 𝒓𝒓 is on the right-hand side of the screen. (b) In the complementary setup, the screen is removed and a thin, perfectly conducting obstacle is placed at the location of the aperture. In addition, the incident 𝑬𝑬 and 𝑩𝑩 fields are changed to 𝑬𝑬 inc ( c ) = 𝑐𝑐𝑩𝑩 inc and 𝑩𝑩 inc ( c ) = −𝑬𝑬 inc 𝑐𝑐⁄ . § The proof of this statement is rather trivial. The incident wave is a superposition of EM plane-waves. For any and all such plane-waves, knowledge of the tangential component of either 𝑬𝑬 or 𝑩𝑩 in the 𝑥𝑥𝑦𝑦 -plane at 𝑧𝑧 = 0 suffices to fully specify the plane-wave in its entirety. It is immaterial whether one specifies 𝑬𝑬 ∥ or 𝑩𝑩 ∥ over the entire plane, or use a patchwork wherein 𝑬𝑬 ∥ is specified in some areas and 𝑩𝑩 ∥ in the remaining areas of the 𝑥𝑥𝑦𝑦 -plane. Given that 𝓔𝓔 ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) equals 𝑬𝑬 inc ∥ on the metallic surface in Fig.K1(a), while 𝓑𝓑 ∥ ( 𝑥𝑥 , 𝑦𝑦 , 0) equals 𝑩𝑩 inc ∥ within the aperture, it is inevitable that ( 𝓔𝓔 , 𝓑𝓑 ) will coincide with ( 𝑬𝑬 inc , 𝑩𝑩 inc ) everywhere in the 𝑧𝑧 ≥ half-space. 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝒓𝒓 × 𝑬𝑬 inc 𝑩𝑩 inc 𝑦𝑦 𝑥𝑥 𝑧𝑧 𝒓𝒓 × 𝑬𝑬 inc ( c ) 𝑩𝑩 inc ( c ) (a) (b) 1 Equations (K2) and (K3) embody the rigorous version of Babinet’s principle of complementary screens for EM waves. Manipulating these equations can yield useful formulas for practical applications. For instance, replacing 𝑬𝑬 inc in Eq.(K2) with its equal −𝑐𝑐𝑩𝑩 inc ( c ) results in 𝑬𝑬 ( 𝒓𝒓 ) = 𝑐𝑐𝑩𝑩 ( c ) ( 𝒓𝒓 ) − 𝑐𝑐𝑩𝑩 inc ( c ) ( 𝒓𝒓 ) = 𝑐𝑐𝑩𝑩 𝑠𝑠 ( c ) ( 𝒓𝒓 ) . (K4) Here, 𝑩𝑩 𝑠𝑠 ( c ) is the scattered 𝐵𝐵 -field in the complementary system of Fig.K1(b). Considering that the tangential component of the scattered 𝐵𝐵 -field vanishes in the open areas of the 𝑥𝑥𝑦𝑦 -plane at 𝑧𝑧 = 0 , the scattered 𝐵𝐵 -field at the observation point can be computed from Eq.(33), as follows: 𝑩𝑩 𝑠𝑠 ( c ) ( 𝒓𝒓 ) = (2 𝜋𝜋 ) −1 𝜵𝜵 × � [ 𝒛𝒛� × 𝑩𝑩 𝑠𝑠 ( c ) ( 𝒓𝒓 )] 𝐺𝐺 ( 𝒓𝒓 , 𝒓𝒓 )d 𝑠𝑠 obstacle . (K5) Thus, Eqs.(K4) and (K5) provide an alternative to Eq.(K1) for computing the diffracted 𝐸𝐸 -field that passes through the aperture in Fig.K1(a). Appendix L
To compute the scattering cross-section of a small dielectric sphere, let the particle of small radius 𝑅𝑅 and isotropic dielectric constant 𝜀𝜀 𝜀𝜀 ( 𝜔𝜔 ) = 𝜀𝜀 [1 + 𝜒𝜒 e ( 𝜔𝜔 )] be centered at the origin of the coordinate system and excited by the incident plane-wave 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑬𝑬 inc exp[i 𝑘𝑘 ( 𝝈𝝈 inc ∙ 𝒓𝒓 − 𝑐𝑐𝑡𝑡 )] . In the static approximation, the induced polarization 𝑷𝑷 𝑒𝑒 −i𝜔𝜔𝑡𝑡 of the dipole produces the uniform self-field 𝑬𝑬 self ( 𝑡𝑡 ) = − ( 𝑷𝑷 𝜀𝜀 ⁄ ) 𝑒𝑒 −i𝜔𝜔𝑡𝑡 , which, in combination with the incident 𝐸𝐸 -field, yields 𝑷𝑷 = 𝜀𝜀 𝜒𝜒 e ( 𝜔𝜔 ) �𝑬𝑬 inc − 𝑷𝑷 � → 𝑷𝑷 = 𝜀𝜀 � e ( 𝜔𝜔 ) 𝜒𝜒 e ( 𝜔𝜔 ) + � 𝑬𝑬 inc = 3 𝜀𝜀 � 𝜀𝜀 ( 𝜔𝜔 ) − ( 𝜔𝜔 ) + � 𝑬𝑬 inc . (L1) The induced dipole moment of the particle is thus 𝒑𝒑 ( 𝑡𝑡 ) = 𝒑𝒑 𝑒𝑒 −i𝜔𝜔𝑡𝑡 , where 𝒑𝒑 = (4 𝜋𝜋 ⁄ ) 𝑅𝑅 𝑷𝑷 = 4 𝜋𝜋𝜀𝜀 𝑅𝑅 � 𝜀𝜀 ( 𝜔𝜔 ) − ( 𝜔𝜔 ) + � 𝑬𝑬 inc . (L2) If the dipole happens to be aligned with the 𝑧𝑧 -axis, then at the far away observation point 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 the radiated 𝐸𝐸 -field in the ( 𝑟𝑟 , 𝜃𝜃 , 𝜑𝜑 ) spherical coordinate system will be 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = − 𝑝𝑝 𝑘𝑘 ( 𝜔𝜔 𝑐𝑐⁄ ) sin 𝜃𝜃 𝑒𝑒 −i𝜔𝜔 ( 𝑡𝑡−𝑘𝑘 𝑐𝑐⁄ ) 𝜽𝜽� . (L3) In vector notation, we can replace 𝑝𝑝 sin 𝜃𝜃 𝜽𝜽� with the triple cross product ( 𝒑𝒑 × 𝝈𝝈 ) × 𝝈𝝈 , which no longer restricts 𝒑𝒑 to the direction of the 𝑧𝑧 -axis. The far-field 𝑬𝑬 and 𝑯𝑯 are now written 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = − ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑘𝑘 𝑒𝑒 −i𝜔𝜔 ( 𝑡𝑡−𝑘𝑘 𝑐𝑐⁄ ) ( 𝒑𝒑 × 𝝈𝝈 ) × 𝝈𝝈 . (L4) 𝑯𝑯 ( 𝒓𝒓 , 𝑡𝑡 ) = − ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑍𝑍 𝑘𝑘 𝑒𝑒 −i𝜔𝜔 ( 𝑡𝑡−𝑘𝑘 𝑐𝑐⁄ ) 𝒑𝒑 × 𝝈𝝈 . (L5) The time-averaged Poynting vector at the observation point is found to be 〈𝑺𝑺 ( 𝒓𝒓 , 𝑡𝑡 ) 〉 = ½Re( 𝑬𝑬 × 𝑯𝑯∗ ) = 𝜇𝜇 𝜔𝜔 𝑐𝑐𝑘𝑘 [( 𝒑𝒑 ∗ × 𝝈𝝈 ) ∙ ( 𝒑𝒑 × 𝝈𝝈 )] 𝝈𝝈 = 𝜇𝜇 𝜔𝜔 𝑐𝑐𝑘𝑘 [ 𝒑𝒑 ∙ 𝒑𝒑 ∗ − ( 𝒑𝒑 ∙ 𝝈𝝈 )( 𝒑𝒑 ∗ ∙ 𝝈𝝈 )] 𝝈𝝈 = ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑅𝑅 𝑘𝑘 � 𝜀𝜀 ( 𝜔𝜔 ) − ( 𝜔𝜔 ) + � [ 𝑬𝑬 inc ∙ 𝑬𝑬 inc ∗ − ( 𝑬𝑬 inc ∙ 𝝈𝝈 )( 𝑬𝑬 inc ∗ ∙ 𝝈𝝈 )] 𝝈𝝈 . (L6) 𝒂𝒂 × ( 𝒃𝒃 × 𝒄𝒄 ) = ( 𝒂𝒂 ∙ 𝒄𝒄 ) 𝒃𝒃 − ( 𝒂𝒂 ∙ 𝒃𝒃 ) 𝒄𝒄 ( 𝒂𝒂 × 𝒃𝒃 ) ∙ ( 𝒄𝒄 × 𝒅𝒅 ) = ( 𝒂𝒂 ∙ 𝒄𝒄 )( 𝒃𝒃 ∙ 𝒅𝒅 ) − ( 𝒂𝒂 ∙ 𝒅𝒅 )( 𝒃𝒃 ∙ 𝒄𝒄 ) 𝒓𝒓 is 𝑟𝑟 d Ω = 𝑟𝑟 sin 𝜃𝜃 d 𝜃𝜃 d 𝜑𝜑 , the EM energy per unit time, d ℰ , crossing this surface will satisfy the following equation: dℰdΩ = ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑅𝑅 � 𝜀𝜀 ( 𝜔𝜔 ) − ( 𝜔𝜔 ) + � � 𝑬𝑬 inc ∙ 𝑬𝑬 inc ∗ − ( 𝑬𝑬 inc ∙ 𝝈𝝈 )( 𝑬𝑬 inc ∗ ∙ 𝝈𝝈 ) � . (L7) In general, the time-averaged incident optical energy per unit time per unit cross-sectional area of the incident plane-wave is ℰ inc = ½Re( 𝑬𝑬 inc × 𝑯𝑯 inc ∗ ) = ½Re[ 𝑬𝑬 inc × ( 𝝈𝝈 inc × 𝑬𝑬 inc ∗ ) 𝑍𝑍 ⁄ ] = ( 𝑬𝑬 inc ∙ 𝑬𝑬 inc ∗ 𝑍𝑍 ⁄ ) 𝝈𝝈 inc . (L8) Consequently, the differential scattering cross section of the particle is d𝒮𝒮dΩ = ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑅𝑅 � 𝜀𝜀 ( 𝜔𝜔 ) − ( 𝜔𝜔 ) + � � − | 𝑬𝑬 inc ∙ 𝝈𝝈 | | 𝑬𝑬 inc | � . (L9) To find the total scattering cross section, we must integrate the above expression over the surface of a unit sphere centered at the origin of coordinates. The first term integrates to ∫ d Ω = ∫ ∫ sin 𝜃𝜃 d 𝜃𝜃 d 𝜑𝜑 = 4 𝜋𝜋 . As for the second term, writing 𝑬𝑬 inc = 𝐸𝐸 𝑥𝑥 𝒙𝒙� + 𝐸𝐸 𝑦𝑦 𝒚𝒚� + 𝐸𝐸 𝑧𝑧 𝒛𝒛� and 𝝈𝝈 = sin 𝜃𝜃 cos 𝜑𝜑 𝒙𝒙� + sin 𝜃𝜃 sin 𝜑𝜑 𝒚𝒚� + cos 𝜃𝜃 𝒛𝒛� , we will have | 𝑬𝑬 inc ∙ 𝝈𝝈 | = ( 𝐸𝐸 𝑥𝑥 sin 𝜃𝜃 cos 𝜑𝜑 + 𝐸𝐸 𝑦𝑦 sin 𝜃𝜃 sin 𝜑𝜑 + 𝐸𝐸 𝑧𝑧 cos 𝜃𝜃 ) ∙ ( 𝐸𝐸 𝑥𝑥∗ sin 𝜃𝜃 cos 𝜑𝜑 + 𝐸𝐸 𝑦𝑦∗ sin 𝜃𝜃 sin 𝜑𝜑 + 𝐸𝐸 𝑧𝑧∗ cos 𝜃𝜃 ) = | 𝐸𝐸 𝑥𝑥 | sin 𝜃𝜃 cos 𝜑𝜑 + | 𝐸𝐸 𝑦𝑦 | sin 𝜃𝜃 sin 𝜑𝜑 + | 𝐸𝐸 𝑧𝑧 | cos 𝜃𝜃 + ½( 𝐸𝐸 𝑥𝑥 𝐸𝐸 𝑦𝑦∗ + 𝐸𝐸 𝑥𝑥∗ 𝐸𝐸 𝑦𝑦 ) sin 𝜃𝜃 sin(2 𝜑𝜑 ) +½( 𝐸𝐸 𝑥𝑥 𝐸𝐸 𝑧𝑧∗ + 𝐸𝐸 𝑧𝑧 𝐸𝐸 𝑥𝑥∗ ) sin(2 𝜃𝜃 ) cos 𝜑𝜑 + ½( 𝐸𝐸 𝑦𝑦 𝐸𝐸 𝑧𝑧∗ + 𝐸𝐸 𝑧𝑧 𝐸𝐸 𝑦𝑦∗ ) sin(2 𝜃𝜃 ) cos 𝜑𝜑 . (L10) ∫ | 𝑬𝑬 inc ∙ 𝝈𝝈 | d Ω spheresurface = ∫ ∫ | 𝑬𝑬 inc ∙ 𝝈𝝈 | sin 𝜃𝜃 d 𝜃𝜃 d 𝜑𝜑 𝜋𝜋𝜃𝜃=0 = � [ 𝜋𝜋 (| 𝐸𝐸 𝑥𝑥 | + | 𝐸𝐸 𝑦𝑦 | ) sin 𝜃𝜃 + 2 𝜋𝜋 | 𝐸𝐸 𝑧𝑧 | sin 𝜃𝜃 cos 𝜃𝜃 ]d 𝜃𝜃 𝜋𝜋𝜃𝜃=0 = (4 𝜋𝜋 ⁄ )(| 𝐸𝐸 𝑥𝑥 | + | 𝐸𝐸 𝑦𝑦 | + | 𝐸𝐸 𝑧𝑧 | ) = (4 𝜋𝜋 ⁄ )| 𝑬𝑬 inc | . (L11) The total scattering cross-section of the isotropic dielectric sphere is thus seen to be independent of the polarization state of the incident beam and given by 𝒮𝒮 = � � � 𝜀𝜀 ( 𝜔𝜔 ) − ( 𝜔𝜔 ) + � � 𝜔𝜔𝑐𝑐 � 𝑅𝑅 . (L12) For linearly polarized light, if the incident polarization happens to be perpendicular to the plane of incidence (i.e., the plane defined by 𝝈𝝈 inc and 𝝈𝝈 ), then, in the differential cross-section formula of Eq.(L9), we will have 𝑬𝑬 inc ∙ 𝝈𝝈 = 0 . However, for an incident polarization in the plane of incidence, 𝑬𝑬 inc ∙ 𝝈𝝈 = 𝐸𝐸 inc sin 𝜗𝜗 , where 𝜗𝜗 is the deviation angle of 𝝈𝝈 from 𝝈𝝈 inc . In the case of natural light, which is unpolarized, the incident beam is an equal mixture of linear polarizations along the ∥ and ⊥ directions. The degree of polarization of the scattered light is then given by Π ( 𝜗𝜗 ) = ( d𝒮𝒮 ⊥ dΩ⁄ ) − �d𝒮𝒮 ∥ dΩ⁄ � ( d𝒮𝒮 ⊥ dΩ⁄ ) + �d𝒮𝒮 ∥ dΩ⁄ � = − ( − sin 𝜗𝜗 ) ( − sin 𝜗𝜗 ) = sin 𝜗𝜗1+cos 𝜗𝜗 . (L13) 3 Appendix M
To compute the scattering cross-section of a small perfectly conducting sphere, we begin by noting that, inside the perfect conductor, both 𝑬𝑬 and 𝑩𝑩 fields must vanish. Assuming the particle radius 𝑅𝑅 is much smaller than the incident wavelength 𝜆𝜆 = 2 𝜋𝜋𝑐𝑐 𝜔𝜔⁄ , and that the static approximation is reasonably accurate to describe the internal fields, the spherical particle acquires an electric dipole moment 𝒑𝒑 𝑒𝑒 −i𝜔𝜔𝑡𝑡 along the direction of the incident 𝐸𝐸 -field, and a magnetic dipole moment 𝒎𝒎 𝑒𝑒 −i𝜔𝜔𝑡𝑡 along the direction of the incident 𝐵𝐵 -field. Denoting the polarization and magnetization of the particle by 𝑷𝑷 and 𝑴𝑴 , respectively, the vanishing of the internal 𝐸𝐸 -field demands that 𝑬𝑬 inc = 𝑷𝑷 𝜀𝜀 ⁄ , while the vanishing of the internal 𝐵𝐵 -field requires that 𝑩𝑩 inc = − 𝑴𝑴 ⁄ . ** Consequently, 𝒑𝒑 = 4 𝜋𝜋𝜀𝜀 𝑅𝑅 𝑬𝑬 inc . (M1) 𝒎𝒎 = − 𝜋𝜋𝜇𝜇 𝑅𝑅 𝑯𝑯 inc . (M2) If 𝒎𝒎 happens to be aligned with the 𝑧𝑧 -axis, then at the far away observation point 𝒓𝒓 = 𝑟𝑟 𝝈𝝈 the radiated 𝐸𝐸 -field in the ( 𝑟𝑟 , 𝜃𝜃 , 𝜑𝜑 ) spherical coordinate system will be (Ref.[11], Problem 4.17) 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑚𝑚 𝜔𝜔 sin 𝜃𝜃 𝑒𝑒 −i𝜔𝜔 ( 𝑡𝑡−𝑘𝑘 𝑐𝑐⁄ ) 𝝋𝝋� . (M3) In vector notation, we can replace 𝑚𝑚 sin 𝜃𝜃 𝝋𝝋� with the cross-product 𝒎𝒎 × 𝝈𝝈 , which no longer restricts 𝒎𝒎 to being aligned with the 𝑧𝑧 -axis. The far-fields 𝑬𝑬 and 𝑯𝑯 will now be written 𝑬𝑬 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜔𝜔 𝑒𝑒 −i𝜔𝜔 ( 𝑡𝑡−𝑘𝑘 𝑐𝑐⁄ ) 𝒎𝒎 × 𝝈𝝈 . (M4) 𝑯𝑯 ( 𝒓𝒓 , 𝑡𝑡 ) = 𝜔𝜔 𝑘𝑘 𝑒𝑒 −i𝜔𝜔 ( 𝑡𝑡−𝑘𝑘 𝑐𝑐⁄ ) 𝝈𝝈 × ( 𝒎𝒎 × 𝝈𝝈 ) . (M5) The total scattered fields are now found by coherent superposition of the scattered fields of the induced electric and magnetic dipoles given by Eqs.(L4), (L5), (M4), and (M5), as follows: 𝑬𝑬 total ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑘𝑘 𝝈𝝈 × [ 𝑬𝑬 inc × ( 𝝈𝝈 − ½ 𝝈𝝈 inc )] 𝑒𝑒 −i𝜔𝜔 ( 𝑡𝑡−𝑘𝑘 𝑐𝑐⁄ ) . (M6) 𝑯𝑯 total ( 𝒓𝒓 , 𝑡𝑡 ) = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑍𝑍 𝑘𝑘 𝝈𝝈 × [ 𝑬𝑬 inc + ½( 𝑬𝑬 inc × 𝝈𝝈 inc ) × 𝝈𝝈 ] 𝑒𝑒 −i𝜔𝜔 ( 𝑡𝑡−𝑘𝑘 𝑐𝑐⁄ ) . (M7) 〈𝑺𝑺 total ( 𝒓𝒓 , 𝑡𝑡 ) 〉 = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑘𝑘 Re { 𝝈𝝈 × [ 𝑬𝑬 inc × ( 𝝈𝝈 − ½ 𝝈𝝈 inc )]} × { 𝝈𝝈 × [ 𝑬𝑬 inc ∗ + ½( 𝑬𝑬 inc ∗ × 𝝈𝝈 inc ) × 𝝈𝝈 ]} = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑘𝑘 Re � { 𝝈𝝈 × [ 𝑬𝑬 inc × ( 𝝈𝝈 − ½ 𝝈𝝈 inc )]} ∙ [ 𝑬𝑬 inc ∗ + ½( 𝑬𝑬 inc ∗ × 𝝈𝝈 inc ) × 𝝈𝝈 ] �𝝈𝝈 ** In the
𝑆𝑆𝑆𝑆 system of units, where 𝑩𝑩 = 𝜇𝜇 𝑯𝑯 + 𝑴𝑴 , the magnitude of a magnetic dipole moment 𝒎𝒎 is 𝜇𝜇 × (circulating current around a loop) × (area of the loop). The magnetization 𝑴𝑴 , being the magnetic dipole moment per unit volume, then has the units of 𝜇𝜇 (i.e., henry/meter) × ampere/meter. This is the same as the units of 𝜇𝜇 𝑯𝑯 , namely, henry × ampere/meter , which is known as weber/meter and often used as the units of 𝑩𝑩 . Our perfectly conducting spherical particle develops surface charges and surface currents in response to the incident plane-wave. The induced surface currents produce the magnetic dipole moment 𝒎𝒎 𝑒𝑒 −i𝜔𝜔𝑡𝑡 , which creates an internal 𝐵𝐵 -field in the same way that the circulating current of a solenoid gives rise to a magnetic 𝐵𝐵 -field inside the solenoid. If the spherical particle were made of a true magnetic material of uniform magnetization, its internal 𝑯𝑯 and 𝑩𝑩 fields would have been −𝑴𝑴 𝜇𝜇 ⁄ and 𝑴𝑴 ⁄ , respectively. However, here we only have a spherical solenoid with an internal 𝐵𝐵 -field of 𝑴𝑴 ⁄ ; the corresponding 𝐻𝐻 -field is simply 𝑴𝑴 𝜇𝜇 ⁄ . = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑘𝑘 Re {[(1 − ½ 𝝈𝝈 ∙ 𝝈𝝈 inc ) 𝑬𝑬 inc − ( 𝝈𝝈 ∙ 𝑬𝑬 inc )( 𝝈𝝈 − ½ 𝝈𝝈 inc )] ∙ [(1 − ½ 𝝈𝝈 ∙ 𝝈𝝈 inc ) 𝑬𝑬 inc ∗ + ½( 𝝈𝝈 ∙ 𝑬𝑬 inc ∗ ) 𝝈𝝈 inc ]} 𝝈𝝈 = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) 𝑘𝑘 [(1 − ½ 𝝈𝝈 ∙ 𝝈𝝈 inc ) 𝑬𝑬 inc ∙ 𝑬𝑬 inc ∗ − ¾( 𝝈𝝈 ∙ 𝑬𝑬 inc )( 𝝈𝝈 ∙ 𝑬𝑬 inc ∗ )] 𝝈𝝈 . (M8) The differential scattering cross section is thus found to be d𝒮𝒮dΩ = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) � (1 − ½ 𝝈𝝈 ∙ 𝝈𝝈 inc ) − | 𝑬𝑬 inc ∙ 𝝈𝝈 | | 𝑬𝑬 inc | � . (M9) For linearly polarized incident beams, differential cross sections for polarization in the plane of incidence ( ∥ ) and perpendicular to the plane of incidence ( ⊥ ) are found from Eq.(M9) to be d𝒮𝒮 ∥ dΩ = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) (cos 𝜗𝜗 − ½) . (M10) d𝒮𝒮 ⊥ dΩ = 𝑅𝑅 ( 𝜔𝜔 𝑐𝑐⁄ ) (1 − ½ cos 𝜗𝜗 ) . (M11) Here, as before, 𝜗𝜗 is the deviation angle of 𝝈𝝈 from 𝝈𝝈 incinc